diff --git a/题库0.3/Problems.json b/题库0.3/Problems.json index 0f55fe07..c9ca6325 100644 --- a/题库0.3/Problems.json +++ b/题库0.3/Problems.json @@ -5665,7 +5665,8 @@ "K0803002B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "$\\dfrac 25$", @@ -5695,7 +5696,8 @@ "K0803002B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "$\\dfrac 3{10}$", @@ -5725,7 +5727,8 @@ "K0803002B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -5749,7 +5752,8 @@ "K0818001X" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -5772,7 +5776,8 @@ "K0803002B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "$\\dfrac 3{10}$", @@ -5804,7 +5809,8 @@ "K0802006B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "(1) $\\Omega = \\{W_1B_1,W_1B_2,W_2B_1,W_2B_2,W_3B_1,W_3B_2,B_1B_2\\}$; (2) $\\dfrac 3{10}$; (3) $\\dfrac 35$", @@ -5834,7 +5840,8 @@ "K0807003B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "$0.95$", @@ -5866,7 +5873,8 @@ "K0818002X" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -5890,7 +5898,8 @@ "K0808002B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "$\\dfrac 9{16}$", @@ -5920,7 +5929,8 @@ "K0806004B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "$\\dfrac 23$", @@ -5950,7 +5960,8 @@ "K0803002B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -5974,7 +5985,8 @@ "K0806004B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "$\\dfrac {13}{42}$", @@ -6004,7 +6016,8 @@ "K0807003B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "$6\\%$", @@ -6038,7 +6051,8 @@ "K0806004B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -8792,7 +8806,9 @@ "K0803002B" ], "tags": [ - "第八单元" + "第八单元", + "概率", + "加法原理与乘法原理" ], "genre": "填空题", "ans": "$\\dfrac 1{108}$", @@ -9259,7 +9275,8 @@ "content": "将$6$辆不同的小汽车和$2$辆不同的卡车驶入如图所示的$10$个车位中的某$8$个内, 其中$2$辆卡车必须停在$A$与$B$的位置, 那么不同的停车位置安排共有\\blank{50}种(结果用数值表示).\n\\begin{center}\n \\begin{tikzpicture}[>=latex]\n \\draw (0,0) node {$B$};\n \\draw (0,1.2) node {$A$};\n \\foreach \\i in {-0.3,0.3,0.9,1.5}{\\draw (-0.3,\\i) -- (2.7,\\i);};\n \\foreach \\i in {-0.3,0.3,...,2.8}{\\draw (\\i,-0.3) -- (\\i, 0.3) (\\i, 0.9) -- (\\i, 1.5);};\n \\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "填空题", "ans": "$40320$", @@ -9473,7 +9490,8 @@ "content": "甲、乙两人从$5$门不同的选修课中各选修$2$门, 则甲、乙所选的课程中恰有$1$门相同的选法有\\blank{50}种.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "填空题", "ans": "$60$", @@ -9744,7 +9762,8 @@ "content": "若$(1+x)^5=a_0+a_1x+a_2x^2+\\cdots+a_5x^5$, 则$a_1+a_2+\\cdots+a_5=$\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "二项式定理" ], "genre": "填空题", "ans": "$31$", @@ -10335,7 +10354,9 @@ "content": "将序号分别为1、2、3、4、5的$5$张参观券全部分给$4$人, 每人至少一张, 如果分给同一人的$2$张参观券连号, 那么不同的分法种数是\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合", + "排列" ], "genre": "填空题", "ans": "$96$", @@ -10662,7 +10683,9 @@ "K0806004B" ], "tags": [ - "第八单元" + "第八单元", + "概率", + "组合" ], "genre": "填空题", "ans": "$\\frac 57$", @@ -10937,7 +10960,9 @@ "K0803002B" ], "tags": [ - "第八单元" + "第八单元", + "概率", + "排列" ], "genre": "填空题", "ans": "$\\frac 14$", @@ -11007,7 +11032,8 @@ "content": "设$(1+x)^n=a_0+a_1x+a_2x^2+a_3x^3+\\cdots +a_nx^n$, 若$\\dfrac{a_2}{a_3}=\\dfrac13$, 则$n=$\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "二项式定理" ], "genre": "填空题", "ans": "$11$", @@ -11564,7 +11590,8 @@ "content": "从单词``shadow''中任意选取$4$个不同的字母排成一排, 则其中含有``a''的共有\\blank{50}种排法(用数字作答).", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "填空题", "ans": "$240$", @@ -11785,7 +11812,9 @@ "K0803002B" ], "tags": [ - "第八单元" + "第八单元", + "组合", + "排列" ], "genre": "填空题", "ans": "$\\frac 25$", @@ -12411,7 +12440,8 @@ "content": "某班班会准备从含甲、乙的$6$名学生中选取$4$人发言, 要求甲、乙两人至少有一人参加, 那么不同的发言顺序有\\blank{50}种.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "填空题", "ans": "$336$", @@ -12647,7 +12677,8 @@ "content": "从$5$名学生中任选$3$人分别担任语文、数学、英语课代表, 其中学生甲不能担任数学课代表, 共有\\blank{50}种不同的选法(结果用数值表示).", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "填空题", "ans": "$48$", @@ -13335,7 +13366,8 @@ "content": "用$1,2,3,4,5$共$5$个数排成一个没有重复数字的三位数, 则这样的三位数有\\blank{50}个.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "填空题", "ans": "$60$", @@ -13687,7 +13719,9 @@ "K0816004X" ], "tags": [ - "第八单元" + "第八单元", + "组合", + "加法原理与乘法原理" ], "genre": "填空题", "ans": "$18$", @@ -13997,7 +14031,8 @@ "content": "某高级中学欲从本校的$7$位古诗词爱好者(其中男生$2$人、女生$5$人)中随机选取$3$名同学作为学校诗词朗读比赛的主持人. 若要求主持人中至少有一位是男同学, 则不同选取方法的种数是\\blank{50}(结果用数值表示).", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "填空题", "ans": "$25$", @@ -14242,7 +14277,8 @@ "K0806004B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "$\\dfrac 7{26}$", @@ -14458,7 +14494,8 @@ "content": "从$5$名志愿者中选出$3$名, 分别从事布置、迎宾、策划三项不同的工作, 每人承担一项工作, 则不同的选派方案共有\\blank{50}种(结果用数值表示).", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "填空题", "ans": "$60$种", @@ -14768,7 +14805,8 @@ "content": "从$4$名男同学和$6$名女同学中选取$3$人参加某社团活动, 选出的$3$人中男女同学都有的不同选法种数是\\blank{50}(用数字作答).", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "填空题", "ans": "$96$", @@ -15118,7 +15156,8 @@ "content": "设$a_1,a_2,a_3,a_4$是$1,2,3,4$的一个排列, 若至少有一个$i\\ (i=1,2,3,4)$使得$a_i=i$成立, 则满足此条件的不同排列的个数为\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "加法原理与乘法原理" ], "genre": "填空题", "ans": "$15$", @@ -15634,7 +15673,8 @@ "K0806004B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "$\\frac{31}{36}$", @@ -16103,7 +16143,8 @@ "K0803002B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "$1$", @@ -16628,7 +16669,8 @@ "K0803002B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "$\\frac 7{10}$", @@ -16891,7 +16933,8 @@ "K0808002B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "$\\frac 1{169}$", @@ -17121,7 +17164,8 @@ "content": "若$(x+2)^n=x^n+ax^{n-1}+\\cdots+bx+c \\ (n\\in \\mathbf{N}^*, \\ n\\ge 3)$, 且$b=4c$, 则$a$的值为\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "二项式定理" ], "genre": "填空题", "ans": "$16$", @@ -17229,7 +17273,8 @@ "K0808002B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "$\\frac 29$", @@ -17524,7 +17569,8 @@ "content": "甲与其四位朋友各有一辆私家车, 甲的车牌尾数是$0$, 其四位朋友的车牌尾数分别是$0$, $2$, $1$, $5$, 为遵守当地4月1日至5日$5$天的限行规定(奇数日车牌尾数为奇数的车通行, 偶数日车牌尾数为偶数的车通行), 五人商议拼车出行, 每天任选一辆符合规定的车, 但甲的车最多只能用一天, 则不同的用车方案总数为\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "填空题", "ans": "$64$", @@ -18036,7 +18082,8 @@ "K0803002B" ], "tags": [ - "第八单元" + "第八单元", + "加法原理与乘法原理" ], "genre": "填空题", "ans": "$\\dfrac 37$", @@ -18178,7 +18225,8 @@ "K0803002B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "$\\dfrac 35$", @@ -18512,7 +18560,8 @@ "K0806004B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "$0.98$", @@ -18876,7 +18925,8 @@ "K0808002B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "$\\frac{13}{15}$", @@ -19156,7 +19206,8 @@ "K0808002B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "$0.03$", @@ -19387,7 +19438,8 @@ "K0808002B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "$\\dfrac 79$", @@ -19611,7 +19663,8 @@ "content": "在报名的$8$名男生和$5$名女生中, 选取$6$人参加志愿者活动, 要求男、女生都有, 则不同的选取方式的种数为\\blank{50}(结果用数值表示).", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "填空题", "ans": "$1688$", @@ -19669,7 +19722,8 @@ "K0806004B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "$\\frac 3{10}$", @@ -20467,7 +20521,8 @@ "K0803002B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "$\\frac 16$", @@ -20701,7 +20756,8 @@ "content": "若将函数$f(x)=x^6$表示成$f(x)=a_0+a_1(x-1)+a_2(x-1)^2+a_3(x-1)^3+\\cdots+a_6(x-1)^6$则$a_3$的值等于\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "二项式定理" ], "genre": "填空题", "ans": "$20$", @@ -21029,7 +21085,8 @@ "content": "将一枚质地均匀的硬币连续抛掷$5$次, 则恰好有$3$次出现正面向上的概率是\\blank{50}(结果用数值表示).", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "$\\frac 5{16}$", @@ -21248,7 +21305,8 @@ "K0803002B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "$0.6$", @@ -21415,7 +21473,8 @@ "K0803002B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "$\\frac{1}2$", @@ -22243,7 +22302,8 @@ "content": "书架上有上、中、下三册的《白话史记》和上、下两册的《古诗文鉴赏辞典》, 现将这五本书从左到右摆放在一起, 则中间位置摆放中册《白话史记》的不同摆放种数为\\blank{50}(结果用数值表示).", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "填空题", "ans": "$24$", @@ -22316,7 +22376,8 @@ "K0808002B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "$\\frac 2{21}$", @@ -22788,7 +22849,8 @@ "K0803002B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "$\\frac 12$", @@ -23177,7 +23239,8 @@ "K0803002B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "$\\frac 7{16}$", @@ -24053,7 +24116,9 @@ "content": "现有$5$位教师要带$3$个班级外出参加志愿者服务, 要求每个班级至多两位老师带队, 且教师甲、乙不能单独带队, 则不同的带队方案有\\blank{50}(用数字作答).", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合", + "加法原理与乘法原理" ], "genre": "填空题", "ans": "$54$", @@ -24376,7 +24441,8 @@ "content": "从集合$A=\\{1,2,3,4,5,6,7,8,9,10\\}$中任取两个数, 欲使取到的一个数大于$k$, 另一个数小于$k$(其中$k\\in A$)的概率是$\\dfrac25$, 则$k=$\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "$4$或$7$", @@ -24701,7 +24767,8 @@ "content": "在报名的$5$名男生和$4$名女生中, 选取$5$人参加志愿者服务, 要求男、女生都有, 则不同的选取方式的种数为\\blank{50}(结果用数值表示).", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "填空题", "ans": "$125$", @@ -25369,7 +25436,8 @@ "content": "从$6$名男医生和$3$名女医生中选出$5$人组成一个医疗小组, 若这个小组中必须男女医生都有, 共有\\blank{50}种不同的组建方案(结果用数值表示).", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "填空题", "ans": "$120$", @@ -26476,7 +26544,8 @@ "K0803002B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "$\\dfrac{11}{12}$", @@ -68190,7 +68259,8 @@ "content": "某外语组有$9$人, 每人至少会英语和日语中的一门, 其中$7$人会英语, $3$人会日语, 从中选出会英语与日语的各一人, 有\\blank{80}种不同的选法.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "加法原理与乘法原理" ], "genre": "填空题", "ans": "", @@ -68214,7 +68284,8 @@ "content": "三位同学分别从``物理拓展''和``化学拓展''这两门课程中选修一门或两门课程, 不同的选法有\\blank{80}种.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "加法原理与乘法原理" ], "genre": "填空题", "ans": "", @@ -68238,7 +68309,8 @@ "content": "若自然数$x,y$满足$x+y\\le 6$, 则有序自然数对$(x,y)$共有\\blank{80}对.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "加法原理与乘法原理" ], "genre": "填空题", "ans": "", @@ -68262,7 +68334,8 @@ "content": "由$1,2,3,4,5$这五个数字可以组成\\blank{80}个四位数(各位上的数字允许重复).", "objs": [], "tags": [ - "第八单元" + "第八单元", + "加法原理与乘法原理" ], "genre": "填空题", "ans": "", @@ -68288,7 +68361,8 @@ "content": "由$0,1,2,3,4$这五个数字可以组成\\blank{80}个四位数(各位的数字允许重复).", "objs": [], "tags": [ - "第八单元" + "第八单元", + "加法原理与乘法原理" ], "genre": "填空题", "ans": "", @@ -68314,7 +68388,8 @@ "content": "有$3$个应届毕业生报名参加五个单位应聘, 每人报且仅报一个单位, 有\\blank{80}种不同报名方法.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "填空题", "ans": "", @@ -68338,7 +68413,8 @@ "content": "$4$封信要投到$3$个信箱, 共有\\blank{80}种不同投法.(允许将信全部或部分投入某一个信箱)", "objs": [], "tags": [ - "第八单元" + "第八单元", + "加法原理与乘法原理" ], "genre": "填空题", "ans": "", @@ -68364,7 +68440,8 @@ "content": "把$10$个苹果分成三堆(不记顺序), 要求每堆至少$1$个, 至多$5$个, 则不同的分法有\\blank{80}种.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "加法原理与乘法原理" ], "genre": "填空题", "ans": "", @@ -68388,7 +68465,9 @@ "content": "将四名教师分配到三个班级去参加活动, 要求每班至少一名的分配方法共有\\blank{80}种.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列", + "加法原理与乘法原理" ], "genre": "填空题", "ans": "", @@ -68412,7 +68491,8 @@ "content": "已知$a\\in\\{-2,-1,0,1,2,3,4\\}$, $b\\in \\{-3,-2,-1,0,1,2,3,4,5\\}$, 则方程$\\dfrac{x^2}{a}+\\dfrac{y^2}{b}=1$表示的不同双曲线共有\\blank{80}条.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "加法原理与乘法原理" ], "genre": "填空题", "ans": "", @@ -68438,7 +68518,8 @@ "content": "$(a_1+a_2+a_3)(b_1+b_2+b_3+b_4)(c_1+c_2)$展开后，共有\\blank{80}项.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "加法原理与乘法原理" ], "genre": "填空题", "ans": "", @@ -68464,7 +68545,8 @@ "content": "用$1,2,3,4,\\cdots,9$这九个数字组成数字不重复的三位数的个数是\\blank{80}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "填空题", "ans": "", @@ -68488,7 +68570,8 @@ "content": "同时抛掷大小不同的两颗骰子, 有\\blank{80}种不同的结果?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "加法原理与乘法原理" ], "genre": "填空题", "ans": "", @@ -68512,7 +68595,9 @@ "content": "用$0,2,4,6,9$这五个数字可以组成数字不重复的五位偶数共\\blank{80}个.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列", + "加法原理与乘法原理" ], "genre": "填空题", "ans": "", @@ -68536,7 +68621,9 @@ "content": "在由数字$1,2,3,4,5$组成的数字不重复的五位数中, 小于$50000$的奇数有\\blank{80}个.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列", + "加法原理与乘法原理" ], "genre": "填空题", "ans": "", @@ -68560,7 +68647,8 @@ "content": "从$8$个学生(含学生甲)中选$5$个排成一列, 其中不包含学生甲的排法共有\\blank{80}种.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "填空题", "ans": "", @@ -68586,7 +68674,8 @@ "content": "从$8$个学生(含学生乙)中选$5$个排成一列, 其中包含学生乙的排法共有\\blank{80}种.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "填空题", "ans": "", @@ -68612,7 +68701,8 @@ "content": "从$1,2,3,4,5$五个数字中每次取出三个数字组成没有重复数字的三位数, 所有这样的三位数的各位数字之和为\\blank{80}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "填空题", "ans": "", @@ -68636,7 +68726,8 @@ "content": "七人并坐, 甲不坐在最左边, 乙不坐在最右边, 共有\\blank{80}种不同的坐法.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "填空题", "ans": "", @@ -68660,7 +68751,8 @@ "content": "在$0,1,2,3,4,5$这六个数字组成的数字不重复的六位数中, 个位数字小于十位数字的有\\blank{80}个.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "填空题", "ans": "", @@ -68684,7 +68776,8 @@ "content": "若复数$a+b\\mathrm{i}$中的$a,b$均可分别取$0,1,2,\\cdots,9$这$10$个数字中的任一个,那么可以组成不同虚数的个数为\\blank{80}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "加法原理与乘法原理" ], "genre": "填空题", "ans": "", @@ -68708,7 +68801,8 @@ "content": "已知集合$M=\\{a_1,a_2,a_3\\}$, $P=\\{b_1,b_2,b_3,b_4,b_5,b_6\\}$, 若$M$中的不同元素对应到$P$中的像不同, 则这样的映射的个数共有\\blank{80}个.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "填空题", "ans": "", @@ -68734,7 +68828,8 @@ "content": "有红, 黄, 绿三种颜色的信号弹各一粒, 按不同的顺序向天空连发三枪表示不同的信号. 则一共可以发出\\blank{80}种不同的信号.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "填空题", "ans": "", @@ -68760,7 +68855,8 @@ "content": "有红, 黄, 绿三种颜色的信号弹各许多粒, 按不同的顺序向天空连发三枪表示不同的信号. 则一共可以发出\\blank{80}种不同的信号.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "加法原理与乘法原理" ], "genre": "填空题", "ans": "", @@ -68786,7 +68882,8 @@ "content": "将$9$人排成$3$排, 每排$3$人, 要求甲在第二排, 乙与丙在第三排, 则所有的不同排法数为\\blank{80}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "填空题", "ans": "", @@ -68810,7 +68907,8 @@ "content": "记$S=1!+2!+\\cdots+99!$, 则$S$的末两位数字为\\blank{80}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "填空题", "ans": "", @@ -68834,7 +68932,8 @@ "content": "已知$\\mathrm{P}_{56}^{x+6}:\\mathrm{P}_{54}^{x+3}=30800:1$, 则$x=$\\blank{80}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "填空题", "ans": "", @@ -68860,7 +68959,8 @@ "content": "已知$\\mathrm{P}_{2x+1}^4=140\\mathrm{P}_x^3$, 则正整数$x=$\\blank{80}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "填空题", "ans": "", @@ -68884,7 +68984,8 @@ "content": "设$P_n=(n+1)(n+2)\\cdots(n+n)$, $Q_n=1\\cdot 3\\cdot 5\\cdot \\cdots\\cdot(2n-1)$, 则$\\dfrac{P_n}{Q_n}=$\\blank{80}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "填空题", "ans": "", @@ -68908,7 +69009,8 @@ "content": "不等式$2<\\dfrac{(x+1)!}{(x-1)!}\\le 42$的解集为\\blank{80}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "填空题", "ans": "", @@ -68932,7 +69034,8 @@ "content": "已知$\\mathrm{P}_x^5=12\\mathrm{P}_x^3$, 则正整数$x=$\\blank{80}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "填空题", "ans": "", @@ -68956,7 +69059,8 @@ "content": "已知$\\mathrm{P}_n^n+\\mathrm{P}_{n-1}^{n-1}=\\dfrac{1}{5}\\mathrm{P}_{n+1}^{n+1}$, 则$n=$\\blank{80}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "填空题", "ans": "", @@ -68982,7 +69086,8 @@ "content": "某班共有学生$30$人, 每两人之间互通一次电话, 则共打电话\\blank{80}次.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "填空题", "ans": "", @@ -69008,7 +69113,8 @@ "content": "某班共有学生$30$人, 每两人之间互通一份信, 则共写信\\blank{80}封.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "填空题", "ans": "", @@ -69034,7 +69140,9 @@ "content": "$10$个人分乘$3$辆汽车, 要求甲车坐$5$人, 乙车坐$3$人, 丙车坐$2$人, 不同的乘车方法共有\\blank{80}种.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合", + "加法原理与乘法原理" ], "genre": "填空题", "ans": "", @@ -69060,7 +69168,8 @@ "content": "$4$本不同的书分给两个人, 每人两本, 不同的分法共有\\blank{80}种.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "填空题", "ans": "", @@ -69084,7 +69193,9 @@ "content": "高三年级共有$8$个班, 分派$4$个数学教师任教, 每个教师都教两个班, 则不同的分派方式共有\\blank{80}种.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合", + "加法原理与乘法原理" ], "genre": "填空题", "ans": "", @@ -69108,7 +69219,9 @@ "content": "从$1,3,5,7,9$这五个数字中任取$3$个, 从$2,4,6,8$这四个数字中任取$2$个, 能组成数字不重复的五位数共\\blank{80}个.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合", + "排列" ], "genre": "填空题", "ans": "", @@ -69132,7 +69245,9 @@ "content": "从$5$位男同学和$4$位女同学中选出$4$位参加一个座谈会, 要求座谈会的成员中既有男同学, 又有女同学, 有\\blank{80}种不同的选法.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合", + "加法原理与乘法原理" ], "genre": "填空题", "ans": "", @@ -69156,7 +69271,9 @@ "content": "从不同的$4$盆仙人球和$5$盆芦荟中任意取出三盆送人, 要求至少有一盆仙人球, 也至少有一盆芦荟, 则不同的选法共有\\blank{80}种.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合", + "加法原理与乘法原理" ], "genre": "填空题", "ans": "", @@ -69180,7 +69297,9 @@ "content": "$6$张同排连号的电影票, 分给$3$名教师和$3$名学生, 如果要求师生之间相间而坐, 则不同的分法共有\\blank{80}种.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列", + "加法原理与乘法原理" ], "genre": "填空题", "ans": "", @@ -69204,7 +69323,9 @@ "content": "一组$6$条平行线与另一组$3$条平行线互相垂直, 则由它们中的四条围成的矩形的个数为\\blank{80}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合", + "加法原理与乘法原理" ], "genre": "填空题", "ans": "", @@ -69228,7 +69349,9 @@ "content": "在$\\angle AOB$的两边上分别有异于点$O$的$5$个和$6$个点, 以这$12$个点(包括$O$点)为顶点, 共可作出\\blank{80}个不同的三角形.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合", + "加法原理与乘法原理" ], "genre": "填空题", "ans": "", @@ -69252,7 +69375,10 @@ "content": "数$1007$, $1334$, $1531$, $1929$都是以$1$开头的四位数, 且每个数恰好有两个数字相等, 这样的四位数共有\\blank{80}个.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合", + "排列", + "加法原理与乘法原理" ], "genre": "填空题", "ans": "", @@ -69276,7 +69402,8 @@ "content": "化简: $\\dfrac{1}{2!}+\\dfrac{2}{3!}+\\dfrac{3}{4!}+\\cdots+\\dfrac{n}{(n+1)!}=$\\blank{80}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "填空题", "ans": "", @@ -69302,7 +69429,8 @@ "content": "[选做]\n化简: $\\dfrac{3}{1!+2!+3!}+\\dfrac{4}{2!+3!+4!}+\\cdots+\\dfrac{(n+2)}{n!+(n+1)!+(n+2)!}=$\\blank{80}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "填空题", "ans": "", @@ -69328,7 +69456,8 @@ "content": "已知$x$是不小于$3$的正整数, $\\mathrm{C}_x^3:\\mathrm{C}_x^2=44:3$, 则$x=$\\blank{80}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "填空题", "ans": "", @@ -69352,7 +69481,8 @@ "content": "已知$x$是不小于$12$的正整数, $\\mathrm{C}_x^{12}=\\mathrm{C}_x^8$, 则$x=$\\blank{80}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "填空题", "ans": "", @@ -69376,7 +69506,8 @@ "content": "已知$2x,16-x$是不大于$18$的非负整数, $\\mathrm{C}_{18}^{2x}=\\mathrm{C}_{18}^{16-x}$, 则$x=$\\blank{80}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "填空题", "ans": "", @@ -69400,7 +69531,8 @@ "content": "计算: $\\mathrm{C}_m^5-\\mathrm{C}_{m+1}^5+\\mathrm{C}_m^4=$\\blank{80}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "填空题", "ans": "", @@ -69426,7 +69558,8 @@ "content": "不等式$\\mathrm{C}_{21}^{x-4}<\\mathrm{C}_{21}^{x-2}<\\mathrm{C}_{21}^{x-1}$的解集为\\blank{80}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "填空题", "ans": "", @@ -69452,7 +69585,8 @@ "content": "计算: $\\mathrm{C}_2^2+\\mathrm{C}_3^2+\\mathrm{C}_4^2+\\cdot+\\mathrm{C}_{100}^2=$\\blank{80}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "填空题", "ans": "", @@ -69479,7 +69613,8 @@ "content": "计算: $\\mathrm{C}_{97}^{94}+\\mathrm{C}_{97}^{95}+\\mathrm{C}_{98}^{96}+\\mathrm{C}_{99}^{97}=$\\blank{80}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "填空题", "ans": "", @@ -69505,7 +69640,8 @@ "content": "有$6$本不同的书, 分给甲, 乙, 丙三人, 其中甲得$1$本, 乙得$2$本, 丙得$3$本, 共有\\blank{80}种不同的分法.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "填空题", "ans": "", @@ -69535,7 +69671,8 @@ "content": "有$6$本不同的书, 分给甲, 乙, 丙三人, 其中一个人得$1$本, 另一个人得$2$本, 第三个人得$3$本, 共有\\blank{80}种不同的分法.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "填空题", "ans": "", @@ -69565,7 +69702,8 @@ "content": "有$6$本不同的书, 分给甲, 乙, 丙三人, 其中甲得$1$本, 乙得$1$本, 丙得$4$本, 共有\\blank{80}种不同的分法.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "填空题", "ans": "", @@ -69595,7 +69733,8 @@ "content": "有$6$本不同的书, 分给甲, 乙, 丙三人, 其中两人得$1$本, 第三个人得$4$本, 共有\\blank{80}种不同的分法.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "填空题", "ans": "", @@ -69625,7 +69764,8 @@ "content": "有$6$本不同的书, 分给甲, 乙, 丙三人, 其中甲得$2$本, 乙得$2$本, 丙得$2$本, 共有\\blank{80}种不同的分法.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "填空题", "ans": "", @@ -69655,7 +69795,8 @@ "content": "有$6$本不同的书, 分给甲, 乙, 丙三人, 每人得$2$本, 共有\\blank{80}种不同的分法.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "填空题", "ans": "", @@ -69685,7 +69826,8 @@ "content": "已知平面上的九个点$\\{(x,y)||x|\\le1, |y|\\le 1, x,y\\in\\mathbf{Z}\\}$, 以这些点中的三个作为顶点, 能构成\\blank{80}个不同的三角形.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "加法原理与乘法原理" ], "genre": "填空题", "ans": "", @@ -69709,7 +69851,9 @@ "content": "平面内共有$17$个点, 其中有且仅有$5$个点共线, 以这些点中的三个点为顶点的三角形共有\\blank{80}个.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合", + "加法原理与乘法原理" ], "genre": "填空题", "ans": "", @@ -69735,7 +69879,8 @@ "content": "平面内有$7$条不同的直线, 其中有且仅有两条直线平行, 则这七条直线最多(想想为什么要最多)能围成三角形\\blank{80}个.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "填空题", "ans": "", @@ -69761,7 +69906,9 @@ "content": "$M$和$N$是两个不重合的平面, 在平面$M$内取$5$个点, $N$内取$4$个点, 则以这些点为顶点的四面体共有\\blank{80}个.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合", + "加法原理与乘法原理" ], "genre": "填空题", "ans": "", @@ -69785,7 +69932,8 @@ "content": "已知$x$是不大于$7$的非负整数, $\\mathrm{C}_7^x=\\mathrm{C}_7^2$, 则$x=$\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "填空题", "ans": "", @@ -69809,7 +69957,8 @@ "content": "在小于$100000$的正整数中, 含有数字$3$的共有\\blank{50}个.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "加法原理与乘法原理" ], "genre": "填空题", "ans": "", @@ -69833,7 +69982,9 @@ "content": "从$1,2,3,\\cdots,100$中取两数(不计次序)相乘, 其乘积能被$3$整除的取法有\\blank{50}种.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合", + "加法原理与乘法原理" ], "genre": "填空题", "ans": "", @@ -69857,7 +70008,8 @@ "content": "在$1$到$10$的十个自然数中, 任取两个(不计次序)相加所得和为奇数的不同情形共有\\blank{50}种.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "加法原理与乘法原理" ], "genre": "填空题", "ans": "", @@ -69881,7 +70033,9 @@ "content": "由数字$1,2,3,4,5$可以组成没有重复数字的五位数$120$个, 若把这些数从小到大排成一列, 第一个数是$12345$, 那么第$93$个数是\\blank{50}, $43251$是第\\blank{50}个数.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合", + "加法原理与乘法原理" ], "genre": "填空题", "ans": "", @@ -69905,7 +70059,9 @@ "content": "在由$1,2,3,\\cdots,9$这九个数字组成的数字不重复的五位数中, 奇数位上一定是奇数的共有\\blank{50}个.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列", + "加法原理与乘法原理" ], "genre": "填空题", "ans": "", @@ -69931,7 +70087,9 @@ "content": "在由$1,2,3,\\cdots,9$这九个数字组成的数字不重复的五位数中, 奇数数字一定在奇数位上的共有\\blank{50}个.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列", + "加法原理与乘法原理" ], "genre": "填空题", "ans": "", @@ -69957,7 +70115,8 @@ "content": "用$1,2,6,9$四个数字组成的所有各位数字不同的四位数之和为\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "填空题", "ans": "", @@ -69981,7 +70140,8 @@ "content": "在所有的四位数中, 千位, 百位, 十位, 个位依次减小的有\\blank{50}个.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "填空题", "ans": "", @@ -70005,7 +70165,9 @@ "content": "不定方程$x_1+x_2+x_3+x_4+x_5=10$的正整数解有\\blank{50}组, 非负整数解有\\blank{50}组.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合", + "加法原理与乘法原理" ], "genre": "填空题", "ans": "", @@ -70029,7 +70191,9 @@ "content": "三张卡片的正反面分别写有数字$1$和$2$, $3$和$4$, $5$和$7$, 若将三张卡片并列, 可得到\\blank{50}个不同的三位数.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列", + "加法原理与乘法原理" ], "genre": "填空题", "ans": "", @@ -70053,7 +70217,8 @@ "content": "$8$个同学排成一排的排列数为$m$, $8$个同学排成前后两排(前排$3$个, 后排$5$个)的排列数为$n$, 则$m,n$的大小关系为\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "填空题", "ans": "", @@ -70077,7 +70242,8 @@ "content": "联欢会上要演出$4$个歌唱节目和$3$个舞蹈节目, 如果舞蹈节目不能连排, 有\\blank{50}种排节目单的方法.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "填空题", "ans": "", @@ -70103,7 +70269,8 @@ "content": "五本不同的小说和$3$本不同的漫画并排放在书架上, 要求$3$本漫画排在一起, 共有\\blank{50}中排放的方法.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "填空题", "ans": "", @@ -70127,7 +70294,8 @@ "content": "要排一张有$5$个独唱节目和$3$个合唱节目的演出节目表, 如果合唱节目不排在节目表的第一个位置上, 并且任何两个合唱节目不相邻, 则不同的排法总数是\\blank{50}种.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "填空题", "ans": "", @@ -70151,7 +70319,8 @@ "content": "在连续的$6$次射击中, 恰好命中$4$次, 且其中恰好有$3$次是连续命中的情形共有\\blank{50}种.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "填空题", "ans": "", @@ -70175,7 +70344,8 @@ "content": "圆周上有$8$个等分点, 以这$8$个点为顶点作直角三角形, 一共可以作\\blank{50}个.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "加法原理与乘法原理" ], "genre": "填空题", "ans": "", @@ -70199,7 +70369,8 @@ "content": "将$6$件不同的产品分别装入两个相同的口袋里, 要求每袋至少有一件, 则不同的装法共有\\blank{50}种.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "填空题", "ans": "", @@ -70223,7 +70394,8 @@ "content": "楼梯一共有$10$级, 上楼可以一步上一级, 也可以一步上两级, 若要求恰好$8$步走完这个楼梯, 则不同的走法一共有\\blank{50}种.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "填空题", "ans": "", @@ -70247,7 +70419,8 @@ "content": "楼梯一共有$10$级, 上楼可以一步上一级, 也可以一步上两级, \\underline{还可以一步上三级}, 走完这个楼梯的不同的走法一共有\\blank{50} 种.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "加法原理与乘法原理" ], "genre": "填空题", "ans": "", @@ -70271,7 +70444,8 @@ "content": "$5$名运动员参加$100$米决赛, 满足每个人所花时间都不相同, 且甲比乙先到终点的最终排名共有\\blank{50} 种.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "填空题", "ans": "", @@ -70295,7 +70469,8 @@ "content": "一天要排入语文, 数学, 英语, 物理, 化学, 体育六节课(上午四节, 下午两节), 要求上午第一节课不排体育, 语文课排在上午, 数学课排在下午, 有\\blank{50}种不同的排课方法.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "填空题", "ans": "", @@ -70319,7 +70494,8 @@ "content": "取$1,2,3,4,5$这五个数字中的两个分别作为一个对数的底数和真数, 则所得的不同值共有\\blank{50}个.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "填空题", "ans": "", @@ -70345,7 +70521,9 @@ "content": "用$0,1,2,3,4,5$这六个数字, 可以组成\\blank{50}个无重复数字且能被$25$整除的四位数.\n(被$25$整除的整数后两位为$00,25,50,75$.)", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列", + "加法原理与乘法原理" ], "genre": "填空题", "ans": "", @@ -70369,7 +70547,9 @@ "content": "从$5$个高中学生和$4$个初中学生中选$4$个代表, 要求至少有两位高中生和一位初中生, 若这$4$个代表分别到$4$个不同的公司去调查, 则不同的分配方案共有\\blank{50}种.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合", + "排列" ], "genre": "填空题", "ans": "", @@ -70393,7 +70573,9 @@ "content": "由数字$0,1,2,3,4,5$可组成无重复数字的三位奇数共\\blank{50}个.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列", + "加法原理与乘法原理" ], "genre": "填空题", "ans": "", @@ -70417,7 +70599,8 @@ "content": "有翻译$8$人, 其中$3$人只会英语, $2$人只会日语, 其余$3$人既会英语又会日语, 现从中选$6$人, 安排$3$人翻译英语, 另$3$人翻译日语, 则不同的安排方法(不单指选人方法)有\\blank{50}种.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "填空题", "ans": "", @@ -70443,7 +70626,8 @@ "content": "以正方体的顶点为顶点的四面体共有\\blank{50}个.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "加法原理与乘法原理" ], "genre": "填空题", "ans": "", @@ -70467,7 +70651,8 @@ "content": "从$5$个男羽毛球运动员和$4$个女羽毛球运动员中选出四个进行混合双打(男女对男女), 则不同的分组方法(不猜先不挑边)有\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "填空题", "ans": "", @@ -70616,7 +70801,8 @@ "content": "$\\left(x+x^{-1}-1\\right)^5$中的常数项为\\blank{80}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "二项式定理" ], "genre": "填空题", "ans": "", @@ -70898,7 +71084,8 @@ "content": "当$n$是正整数时, $1-2\\mathrm{C}_n^1+4\\mathrm{C}_n^2-8\\mathrm{C}_n^3+\\cdots+(-2)^n\\mathrm{C}_n^n=$\\blank{80}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "二项式定理" ], "genre": "填空题", "ans": "", @@ -70922,7 +71109,8 @@ "content": "求值: $\\mathrm{C}_{100}^{0}-\\mathrm{C}_{100}^2+\\mathrm{C}_{100}^4-\\mathrm{C}_{100}^6+\\cdots-\\mathrm{C}_{100}^{98}+\\mathrm{C}_{100}^{100}=$\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "二项式定理" ], "genre": "填空题", "ans": "", @@ -70996,7 +71184,9 @@ "content": "(1) 求证: $k\\mathrm{C}_n^k=n\\mathrm{C}_{n-1}^{k-1}$.\\\\ \n(2) {\\it (选做)}已知$n$是正整数, 求$\\mathrm{C}_n^0+\\dfrac{1}{2}\\mathrm{C}_n^1+\\dfrac{1}{3}\\mathrm{C}_n^2+\\cdots+\\dfrac{1}{n+1}\\mathrm{C}_n^n$.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "二项式定理", + "组合" ], "genre": "解答题", "ans": "", @@ -71020,7 +71210,8 @@ "content": "在$(x+1)(x+2)(x+3)\\cdots(x+20)$的展开式中, $x^{19}$的系数为\\blank{50}, (选做)$x^{18}$的系数为\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "加法原理与乘法原理" ], "genre": "填空题", "ans": "", @@ -71044,7 +71235,8 @@ "content": "$(a+b+c)^5$合并同类项后共有\\blank{40}项, 其中$a^3bc$的系数为\\blank{40}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "二项式定理" ], "genre": "填空题", "ans": "", @@ -71093,7 +71285,8 @@ "content": "$77^{77}-15$除以$19$的余数为\\blank{40}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "二项式定理" ], "genre": "填空题", "ans": "", @@ -71144,7 +71337,8 @@ "content": "证明: 无论$n$是何正整数, $(n+1)^n-1$能被$n^2$整除.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "二项式定理" ], "genre": "解答题", "ans": "", @@ -71168,7 +71362,8 @@ "content": "模仿下列方式:\\\\ \n``已知$n$是正整数, 证明: $\\mathrm{C}_n^0\\mathrm{C}_n^n+\\mathrm{C}_n^1\\mathrm{C}_n^{n-1}+\\cdots+\\mathrm{C}_n^n\\mathrm{C}_n^0=\\mathrm{C}_{2n}^n$.\\\\ \n证: 假设某班有$n$个男生, $n$个女生. 原式右端可看做在班级的$2n$个人中选$n$个人的选法总数.\\\\ \n而在$2n$个人中选$n$个人有如下的可能:\\\\ \n选$0$个男生, $n$个女生;\\\\ \n选$1$个男生, $(n-1)$个女生;\\\\ \n$\\cdots$\\\\ \n选$n$个男生, $0$个女生;\\\\ \n故选法总数也可以表示成$\\mathrm{C}_n^0\\mathrm{C}_n^n+\\mathrm{C}_n^1\\mathrm{C}_n^{n-1}+\\cdots+\\mathrm{C}_n^n\\mathrm{C}_n^0$. 因此原式成立.''\\\\ \n解决问题: 已知$r,m,n$均为正整数, $r\\le \\min(m,n)$, 则$\\mathrm{C}_m^0\\mathrm{C}_n^r+\\mathrm{C}_m^1\\mathrm{C}_n^{r-1}+\\cdots+\\mathrm{C}_m^r\\mathrm{C}_n^0$的值为\\blank{40}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "填空题", "ans": "", @@ -71219,7 +71414,8 @@ "K0803002B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "", @@ -71245,7 +71441,8 @@ "content": "袋中有$10$个球, 记有号码$0,1,2,3,4,\\cdots,9$, 任意取出$3$个球, 没有号码$3$的概率为\\blank{80}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "", @@ -71271,7 +71468,8 @@ "content": "已知$10$个产品中有$3$个次品, 从中任取$5$个, 则至少有一个次品的概率为\\blank{80}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "", @@ -71297,7 +71495,8 @@ "content": "某种密码由$8$个数字组成, 且每个数字可以是$0,1,2,\\cdots,9$中的任意一个数, 则这种密码由完全不同的数字组成的概率为\\blank{80}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "", @@ -71321,7 +71520,8 @@ "content": "一工厂生产的$10$个产品中有$9$个一等品, $1$个二等品, 现从这批产品中抽取$4$个, 则其中恰好有一个二等品的概率为\\blank{80}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "", @@ -71350,7 +71550,8 @@ "K0806004B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "", @@ -71374,7 +71575,8 @@ "content": "某城镇共有$10000$辆自行车, 牌照编号从$00001$到$10000$. 则在此城镇中偶然遇到一辆自行车, 其牌照号码中有数字$8$的概率为\\blank{80}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "", @@ -71400,7 +71602,8 @@ "content": "某人有$5$把钥匙, 但只有一把能打开门, 他每次取一把钥匙尝试开门, 则试到第$3$把钥匙时才打开门的概率为\\blank{80}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "", @@ -71426,7 +71629,8 @@ "content": "某次测验有$10$道备用试题, 甲同学在这$10$道题中能够答对$6$题. 现在备用试题中随机抽考$5$题, 规定答对$4$题或$5$题为优秀, 则甲同学获优秀的概率为\\blank{80}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "", @@ -71450,7 +71654,8 @@ "content": "从$0,1,2,3,\\cdots,9$这$10$个数字中, 不重复地任取三个数, 则这$3$个数中最小的一个数不大于$5$的概率为\\blank{80}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "", @@ -71476,7 +71681,8 @@ "K0803002B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "", @@ -71504,7 +71710,8 @@ "K0803002B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "", @@ -71530,7 +71737,8 @@ "content": "已知某班有$38$名学生, 小李, 小王, 小张是该班的$3$名学生, 某次班会决定随机地挑选$3$名学生在会上发言. 则小李, 小王, 小张按此次序被选中的概率为\\blank{80}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "", @@ -71556,7 +71764,8 @@ "content": "已知某班有$38$名学生, 小李, 小王, 小张是该班的$3$名学生, 某次班会决定随机地挑选$3$名学生在会上发言. 则小李, 小王, 小张按任意次序被选中的概率为\\blank{80}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "", @@ -71582,7 +71791,8 @@ "content": "一部$4$卷的文集, 按任意次序放到书架上, 则各卷自左向右或自右向左的卷号恰好为$1,2,3,4$的概率为\\blank{80}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "", @@ -71608,7 +71818,8 @@ "content": "一个口袋里装有大小相同的$7$个白球和$3$个黑球, 每个球上都有编号, 而且编号各不相同, 从中任意摸出$3$个球, 则至少有一个是黑球的概率为\\blank{80}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "", @@ -71634,7 +71845,8 @@ "content": "一个口袋里装有大小相同的$7$个白球和$3$个黑球, 球上没有任何记号, 从中任意摸出$3$个球, 则至少有一个是黑球的概率为\\blank{80}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "", @@ -71660,7 +71872,8 @@ "content": "某一批种子, 如果每一粒发芽的概率均为$90\\%$, 现播下$5$粒种子, 其中至少有$2$粒种子发芽的概率为\\blank{80}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "", @@ -71684,7 +71897,8 @@ "content": "某仪表内装有$m$个相同的电子元件, 其中任一个电子元件损坏时, 该仪表就不能正常工作. 如果在某段时间内每个电子元件损坏的概率都是$p$, 则在这段时间内这个仪表能正常工作的概率为\\blank{80}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "", @@ -71708,7 +71922,8 @@ "content": "两个篮球运动员在罚球线投篮的命中率分别是$0.7$和$0.8$, 每人投篮$3$次, 两人都恰好投进$2$次的概率为\\blank{80}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "", @@ -71732,7 +71947,8 @@ "content": "从正方体的八个顶点中任取四个, 能组成四面体的概率是\\blank{80}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "", @@ -71756,7 +71972,8 @@ "content": "八个人排成两排, 每排四人, 甲,乙两人在同排且甲, 乙两人不相邻的概率是\\blank{80}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "", @@ -71782,7 +71999,8 @@ "K0803002B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "$\\dfrac 38$", @@ -71814,7 +72032,8 @@ "K0803002B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "", @@ -71840,7 +72059,8 @@ "K0803002B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "$\\dfrac{16}{33}$", @@ -71870,7 +72090,8 @@ "content": "有两个电路, $A$电路由三个部分串联组成, 其中第一部分有一个元件, 第二部分是两个并联的元件, 第三部分是三个并联的元件; $B$电路由两个部并联组成, 其中第一部分是三个串联的元件, 第二部分也是三个串联的元件. 假如每个元件接通的概率均为$p$.\\\\ \n(1) 画出这两个电路图.\\\\ \n(2) 如果$p=1/2$, 那么哪个电路接通的概率较大?\\\\ \n(3) 如果$p\\in (0,1)$, 哪个电路接通的概率较大? 为什么?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -94826,7 +95047,8 @@ "content": "若$m\\in \\mathbf{N}^*$, $m<27$, 则$(27-m)(28-m)\\cdots (34-m)=$\\blank{50}(用排列数表示).", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "填空题", "ans": "", @@ -94847,7 +95069,8 @@ "content": "$540$的不同的正约数共有\\blank{50}个, 这些正约数中是$3$的倍数的有\\blank{50}个.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "加法原理与乘法原理" ], "genre": "填空题", "ans": "", @@ -94868,7 +95091,9 @@ "content": "$3000$和$8000$之间有\\blank{50}个能被5整除的且在数位上无重复数字的数.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列", + "加法原理与乘法原理" ], "genre": "填空题", "ans": "", @@ -94889,7 +95114,9 @@ "content": "平面上有$10$个点, 其中除有$4$个点在同一条直线上外, 不再有其它三点共线的情形, 经过这些点可以确定\\blank{50}条直线.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合", + "加法原理与乘法原理" ], "genre": "填空题", "ans": "", @@ -94910,7 +95137,8 @@ "content": "从$5$男$3$女共$8$名学生中选出队长$1$人, 副队长$1$人, 普通队员$2$人组成$4$人志愿者服务队, 要求服务队中至少有$1$名女生, 共有\\blank{50}种不同的选法(用数字作答).", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "填空题", "ans": "", @@ -94953,7 +95181,8 @@ "content": "化简: (1) $1+2\\mathrm{C}_n^1+4\\mathrm{C}_n^2+\\cdots+2^n\\mathrm{C}_n^n=$\\blank{50};\\\\\n(2) $\\mathrm{C}_3^3+\\mathrm{C}_4^3+\\mathrm{C}_5^3+\\cdots+\\mathrm{C}_n^3=$\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "二项式定理" ], "genre": "填空题", "ans": "", @@ -94974,7 +95203,8 @@ "content": "随机抽取$9$个同学中, 至少有$2$个同学在同一月出生的概率是\\blank{50}(默认每月天数相同, 结果精确到$0.001$).", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "", @@ -94997,7 +95227,8 @@ "content": "从一副$52$张扑克牌中随机抽取$4$张牌.\\\\\n(1) 在放回抽取的情形下, $4$张牌都是A的概率为\\blank{50};\n(2) 在不放回抽取的情形下, $4$张牌都是A的概率为\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "", @@ -95039,7 +95270,8 @@ "content": "有$3$名男生, $4$名女生, 全体排成一行, 在下列不同的要求下, 分别求不同排列的方法数:\\\\\n(1) 男生必须排在一起;\\\\\n(2) 任意两个男生都不相邻;\\\\\n(3) 甲不在最左边, 乙不在最右边;\\\\\n(4) 甲必须站在乙的左方(不一定相邻);\\\\\n(5) 其中甲、乙、丙三人从左至右的顺序不变(都不一定相邻).", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "解答题", "ans": "", @@ -95103,7 +95335,8 @@ "content": "已知$\\mathrm{C}_{18}^{2x}=\\mathrm{C}_{18}^{x+3}$, 则$x$=\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "填空题", "ans": "", @@ -95127,7 +95360,8 @@ "content": "用$0$、$1$、$2$、$3$、$4$、$5$组成的无重复数字的数中比$240135$大的数有\\blank{50}个.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "填空题", "ans": "", @@ -95148,7 +95382,8 @@ "content": "甲、乙、丙三人值周一至周六的班, 每人值两天班, 若甲不值周一、乙不值周六, 则可排出不同的值班表数为\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "填空题", "ans": "", @@ -95191,7 +95426,8 @@ "content": "设$1+(1+x)+(1+x)^2+\\cdots+(1+x)^{2021}=a_0+a_1x+a_2x^2+\\cdots +a_{2021}x^{2021}$, 则$a_{100}+a_{101}=$\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "二项式定理" ], "genre": "填空题", "ans": "", @@ -95214,7 +95450,8 @@ "K0803002B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "$\\dfrac 3{10}$", @@ -95242,7 +95479,8 @@ "content": "盒子中装着标有$1,2,3,4$的卡片各两张, 从盒中任取$3$张, 每张卡片被抽到的概率相等, 则抽出的$3$张卡片上的数字互不相同的概率为\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "", @@ -95519,7 +95757,8 @@ "K0806004B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "", @@ -96040,7 +96279,8 @@ "content": "从$6$个人选$4$个人去值班, 每人值班一天, 第一天安排$1$个人, 第二天安排$1$个人, 第三天安排$2$个人, 则共有\\blank{50}种安排情况.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "填空题", "ans": "", @@ -96604,7 +96844,8 @@ "content": "某三位数密码, 每位数字可在$0$至$9$这$10$个数字中任选一个, 则该三位数密码中, 恰有两位数字相同的概率是\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "", @@ -97110,7 +97351,8 @@ "K0803002B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "$\\dfrac 15$", @@ -97266,7 +97508,8 @@ "content": "《九章算术》中, 称底面为矩形而有一侧棱垂直于底面的四棱锥为阳马. 设$AA_1$是正六棱柱的一条侧棱, 如图. 若阳马以该正六棱柱的顶点为顶点、以$AA_1$为底面矩形的一边, 则这样的阳马的个数是\\bracket{15}.\n\\begin{center}\n \\begin{tikzpicture}\n \\coordinate (A) at (0,0) node [below] {$A$};\n \\path (A) --++ (45:{sqrt(3)/2}) --++ (0.5,0) coordinate (B);\n \\path (A) --++ (45:{sqrt(3)}) coordinate (C);\n \\path (C) --++ (-1,0) coordinate (D);\n \\path (B) --++ (-2,0) coordinate (E);\n \\coordinate (F) at (-1,0);\n \\draw (F) -- (A) -- (B) -- (C);\n \\draw [dashed] (C) -- (D) -- (E) -- (F);\n \\foreach \\i in {(A),(B),(C),(F)}{\\draw \\i --++ (0,2);};\n \\foreach \\i in {(D),(E)}{\\draw [dashed] \\i --++ (0,2);};\n \\path (A) --++ (0,2) coordinate (A1) node [above] {$A_1$};\n \\path (B) --++ (0,2) coordinate (B1);\n \\path (C) --++ (0,2) coordinate (C1);\n \\path (D) --++ (0,2) coordinate (D1);\n \\path (E) --++ (0,2) coordinate (E1);\n \\path (F) --++ (0,2) coordinate (F1);\n \\draw (A1) -- (B1) -- (C1) -- (D1) -- (E1) -- (F1) -- cycle;\n \\end{tikzpicture}\n\\end{center}\n\\fourch{$4$}{$8$}{$12$}{$16$}", "objs": [], "tags": [ - "第八单元" + "第八单元", + "加法原理与乘法原理" ], "genre": "选择题", "ans": "", @@ -97461,7 +97704,8 @@ "content": "若排列数$\\mathrm{P}_6^m=6\\times 5\\times 4$, 则$m=$\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "填空题", "ans": "", @@ -98771,7 +99015,8 @@ "K0803002B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "", @@ -98934,7 +99179,8 @@ "K0807003B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "", @@ -99326,7 +99572,8 @@ "content": "(理科)设口袋中有黑球、白球共$7$个, 从中任取$2$个球, 已知取到的白球的个数的数学期望为$\\dfrac 6 7$, 则口袋中白球的个数为\\blank{50}.\\\\\n(文科)从$0,1,2,3,4$这五个数中随机取$2$个数组成一个二位数, 则这个二位数为偶数的概率是\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "", @@ -100166,7 +100413,8 @@ "K0803002B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "选择题", "ans": "B", @@ -100616,7 +100864,8 @@ "content": "在一次教师联欢会上, 到会的女教师比男教师多$12$人, 从到会教师中随机挑选一人表演节目. 如果每位教师被选中的概率相等, 而且选中男教师的概率为$\\dfrac 9{20}$, 那么参加这次联欢会的教师共有\\blank{50}人.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "", @@ -100725,7 +100974,8 @@ "K0820001X" ], "tags": [ - "第八单元" + "第八单元", + "二项式定理" ], "genre": "填空题", "ans": "", @@ -100746,7 +100996,8 @@ "content": "若三个人踢毽, 互相传递, 每人每次只能踢以下, 由甲开始踢, 经过$5$次传递后, 毽又被踢回给甲, 则不同的传递方式共有\\blank{50}种.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "加法原理与乘法原理" ], "genre": "填空题", "ans": "", @@ -101039,7 +101290,8 @@ "content": "(理科)有$3$位射击手独立瞄准一个相同目标, 他们命中的概率都是$0.8$, 则目标恰好被两名射手命中的概率是\\blank{50}.\\\\\n(文科)袋子中有大小形状相同的$4$个红球, $3$个白球, 某人随机抽出两个球, 则恰好是一红一白的概率是\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "", @@ -101709,7 +101961,8 @@ "content": "某科技小组有$6$名同学, 现从中选出$3$人去参观展览, 至少有$1$名女生入选时的不同选法有$16$中, 则小组中的女生人数为\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "填空题", "ans": "", @@ -102119,7 +102372,8 @@ "content": "(理科)一个不透明的袋中装有白球、红球共$9$个($9$个球除颜色外其余完全相同), 经充分混合后, 从袋中随机摸出$2$球, 且摸出的$2$球中至少有一个是白球的概率为$\\dfrac 56$, 现用$\\xi$表示摸出的$2$个球中红球的个数, 则随机变量$\\xi$的数学期望$E\\xi=$\\blank{50}.\\\\\n(文科)一个不透明的袋中装有$5$个白球、$4$个红球($9$个球除颜色外其余完全相同), 经充分混合后, 从袋中随机摸出$3$球, 则摸出的$3$球中至少有一个是白球的概率为\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "", @@ -102192,7 +102446,8 @@ "content": "(理科)甲、乙、丙、丁与小强一起比赛象棋, 每两人都要比赛一盘, 到现在为止, 甲已经赛了$4$盘, 乙赛了$3$盘, 丙赛了$2$盘, 丁赛了$1$盘, 则小强已经赛了\\bracket{20}.\n\\fourch{$4$盘}{$3$盘}{$2$盘}{$1$盘}\\\\\n(文科)``$-2\\le a\\le 2$''是``实系数一元二次方程$x^2+ax+1=0$有虚根''的\\bracket{20}.\n\\fourch{充要条件}{必要不充分条件}{充分不必要条件}{既不充分也不必要条件}", "objs": [], "tags": [ - "第八单元" + "第八单元", + "加法原理与乘法原理" ], "genre": "选择题", "ans": "", @@ -102441,7 +102696,8 @@ "K0803002B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "", @@ -103066,7 +103322,8 @@ "K0803002B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "", @@ -103692,7 +103949,8 @@ "content": "计算$1-3\\mathrm{C}_{10}^1+9\\mathrm{C}_{10}^2-27\\mathrm{C}_{10}^3+\\cdots-3^9\\mathrm{C}_{10}^9+3^{10}=$\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "二项式定理" ], "genre": "填空题", "ans": "", @@ -103816,7 +104074,8 @@ "content": "已知$3$名志愿者在$10$月$1$日至$10$月$5$日期间参加$2013$年国庆节志愿者活动工作.\\\\\n(文科)若每名志愿者在$5$天中任选一天参加社区服务工作, 且各志愿者的选择互不影响, 则$3$名志愿者恰好连续$3$天参加社区服务工作的概率为\\blank{50}.\\\\\n(理科)若每名志愿者在这$5$天中任选两天参加社区服务工作, 且各志愿者的选择互不影响, 以$\\xi$表示这$3$名志愿者在$10$月$1$日参加志愿者服务工作的人数, 则随机变量$\\xi$的数学期望为\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "", @@ -104168,7 +104427,8 @@ "content": "设$(1+x)+(1+x)^2+\\cdots+(1+x)^n=a_0+a_1x+a_2x^2+\\cdots+a_nx^n, \\ n\\in \\mathbf{N}^*$, 若$a_1+a_2+\\cdots+a_{n-1}=61-n$, 则$n=$\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "二项式定理" ], "genre": "填空题", "ans": "", @@ -104500,7 +104760,8 @@ "content": "甲、乙、丙三个单位分别需要招聘工作人员$2$名、$1$名、$1$名, 现从$10$名应聘人员中招聘$4$人到甲、乙、丙三个单位, 那么不同的招聘方法共有\\blank{50}种.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "填空题", "ans": "", @@ -104544,7 +104805,8 @@ "content": "某商场开展促销抽奖活动, 摇奖器摇出一组中奖号码为$1,2,3,4,5,6$, 参加抽奖的顾客从$0$至$9$十个号码中任意抽取$6$个, 若其中至少有$5$个号码与摇奖器摇出的号码相同(不计顺序), 则可中奖. 那么某顾客抽一次就能中奖的概率为\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "", @@ -104652,7 +104914,8 @@ "content": "(理科)(1) 甲同学从学校乘车回家, 图中有$3$个交通岗, 假设在各交通岗遇到红灯的时间是相互独立的, 并且概率都是$\\dfrac 25$, 求甲同学回家途中遇到红灯次数的期望值;\\\\\n(2) $A$箱内有$1$个红球和$n+1$个白球, $B$箱内有$n-1$个白球($n\\in \\mathbf{N}, \\ n\\ge 2$), 现随机从$A$箱内取出$3$个球放入$B$箱内, 将$B$箱中的球充分搅拌后, 再从中随机取出$3$个球放入$A$箱, 求红球由$A$箱入$B$箱再返回$A$箱的概率.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -104846,7 +105109,8 @@ "content": "若$(x-2)^5=a_5x^5+a_4x^4+a_3x^3+a_2x^2+a_1x+a_0$, 则$a_1+a_2+a_3+a_4+a_5=$\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "二项式定理" ], "genre": "填空题", "ans": "", @@ -104867,7 +105131,8 @@ "content": "甲、乙、丙$3$位同学选修课程, 从$4$门学科中, 甲选$2$门, 乙、丙各选$3$门, 则不同的选修方案为\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "填空题", "ans": "", @@ -104890,7 +105155,8 @@ "K0803002B" ], "tags": [ - "第八单元" + "第八单元", + "加法原理与乘法原理" ], "genre": "填空题", "ans": "", @@ -105008,7 +105274,8 @@ "content": "甲、乙等$5$位奥运志愿者被分到$A,B,C,D$四个不同的岗位上, 每个岗位至少有一名志愿者, 不同的分配方案概率相灯.\\\\\n(1) 求甲乙两人同时参加$A$岗位服务的概率;\\\\\n(2) (文科)求甲乙两人不在同一岗位服务的概率;\\\\\n(理科)设随机变量$\\xi$表示这$5$名志愿者中参加$A$岗位服务的人数, 求随机变量$\\xi$的概率分布律.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -105425,7 +105692,9 @@ "K0810003X" ], "tags": [ - "第八单元" + "第八单元", + "排列", + "加法原理与乘法原理" ], "genre": "解答题", "ans": "", @@ -105449,7 +105718,8 @@ "K0813004X" ], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "解答题", "ans": "", @@ -105473,7 +105743,8 @@ "K0813004X" ], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "解答题", "ans": "", @@ -105496,7 +105767,8 @@ "K0817006X" ], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "解答题", "ans": "", @@ -105544,7 +105816,8 @@ "K0820001X" ], "tags": [ - "第八单元" + "第八单元", + "二项式定理" ], "genre": "解答题", "ans": "", @@ -105592,7 +105865,8 @@ "K0813004X" ], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "解答题", "ans": "", @@ -105616,7 +105890,8 @@ "K0813004X" ], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "解答题", "ans": "", @@ -105639,7 +105914,9 @@ "K0815003X" ], "tags": [ - "第八单元" + "第八单元", + "组合", + "加法原理与乘法原理" ], "genre": "解答题", "ans": "", @@ -105663,7 +105940,8 @@ "K0813004X" ], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "解答题", "ans": "", @@ -105687,7 +105965,8 @@ "K0811005X" ], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "解答题", "ans": "", @@ -105711,7 +105990,8 @@ "K0820001X" ], "tags": [ - "第八单元" + "第八单元", + "二项式定理" ], "genre": "解答题", "ans": "", @@ -105758,7 +106038,8 @@ "K0811005X" ], "tags": [ - "第八单元" + "第八单元", + "加法原理与乘法原理" ], "genre": "解答题", "ans": "", @@ -105807,7 +106088,8 @@ "K0803002B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -105830,7 +106112,8 @@ "K0821001X" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -105856,7 +106139,8 @@ "K0821003X" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -105881,7 +106165,8 @@ "K0826004X" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -105908,7 +106193,8 @@ "K0826004X" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -105933,7 +106219,8 @@ "K0826004X" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -105956,7 +106243,8 @@ "K0827004X" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -105981,7 +106269,8 @@ "K0828003X" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -106006,7 +106295,8 @@ "K0808002B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -106029,7 +106319,8 @@ "K0821003X" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -106052,7 +106343,8 @@ "K0821003X" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -106077,7 +106369,8 @@ "K0825002X" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -106102,7 +106395,8 @@ "K0825002X" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -106127,7 +106421,8 @@ "K0825002X" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -106150,7 +106445,8 @@ "K0822003X" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -106174,7 +106470,8 @@ "K0822003X" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -106567,7 +106864,8 @@ "content": "平面上有$12$个不同的点, 其中任何$3$点不在同一直线上, 如果任取$3$点作为顶点作三角形, 那么一共可作\\blank{50}个三角形(结果用数值表示).", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "填空题", "ans": "$220$", @@ -107244,7 +107542,8 @@ "K0808002B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "", @@ -107885,7 +108184,8 @@ "K0803002B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "", @@ -108482,7 +108782,9 @@ "content": "设$a,b,c,d,e,f$为$1,2,3,4,5,6$的任意一个排列, 则使得$(a+b)(c+d)(e+f)$为偶数的排列共有\\blank{50}个.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列", + "加法原理与乘法原理" ], "genre": "填空题", "ans": "", @@ -108938,7 +109240,8 @@ "content": "若$(x+2)^n=x^n+ax^{n-1}+\\cdots +bx+c$($n\\in \\mathbf{N}^*$, $ n\\ge 3$), 且$b=4c$, 则$a$的值为\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "二项式定理" ], "genre": "填空题", "ans": "", @@ -108994,7 +109297,8 @@ "K0808002B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "$\\dfrac 29$", @@ -110152,7 +110456,8 @@ "K0803002B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "", @@ -110616,7 +110921,8 @@ "content": "在均匀分布的条件下, 某些概率问题可转化为几何图形的面积比来计算, 勒洛三角形是由德国机械工程专家勒洛首先发现, 作法为: 以等边三角形的每个顶点为圆心, 以边长为半径, 在另两个顶点间作一段弧, 三段弧围成的曲边三角形就是勒洛三角形, 在勒洛三角形中随机取一点, 此点取自正三角形的概率为\\blank{50}.\n\\begin{center}\n \\begin{tikzpicture}\n \\draw (0,0) coordinate (A) -- (2,0) coordinate (B) -- (1,{sqrt(3)}) coordinate (C) -- cycle;\n \\draw (A) arc (240:300:2) arc (0:60:2) arc(120:180:2);\n \\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "", @@ -111159,7 +111465,8 @@ "content": "某班从$4$位男生和$3$位女生志愿者选出$4$人参加校运会的点名签到工作, 则选出的志愿者中既有男生又有女生的概率的是\\blank{50}(结果用最简分数表示).", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "", @@ -111826,7 +112133,8 @@ "content": "上海某高校哲学专业的$4$名研究生到指定的4所高级中学宣讲习近平新时代中国特色社会主义思想. 若他们每人都随机地从$4$所学校选择一所, 则$4$人中至少有$2$人选择到同一所学校的概率是\\blank{50}(结果用最简分数表示).", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "$\\dfrac{29}{32}$", @@ -112878,7 +113186,8 @@ "K0803002B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "", @@ -113612,7 +113921,8 @@ "K0808002B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "选择题", "ans": "D", @@ -114054,7 +114364,8 @@ "K0803002B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "", @@ -114635,7 +114946,9 @@ "content": "记$a,b,c,d,e,f$为$1,2,3,4,5,6$的任意一个排列, 则使得$(a+b)(c+d)(e+f)$为奇数的排列共有\\blank{50}个.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列", + "加法原理与乘法原理" ], "genre": "填空题", "ans": "", @@ -115683,7 +115996,8 @@ "content": "从正方体的$8$个顶点中选取$4$个作为顶点, 可得到四面体的个数为\\bracket{20}.\n\\fourch{$\\mathrm{C}_8^4-12$}{$\\mathrm{C}_8^4-8$}{$\\mathrm{C}_8^4-6$}{$\\mathrm{C}_8^4-4$}", "objs": [], "tags": [ - "第八单元" + "第八单元", + "加法原理与乘法原理" ], "genre": "选择题", "ans": "", @@ -116376,7 +116690,8 @@ "content": "从$5$名志愿者中选出$3$名, 分别从事布置、迎宾、策划三项不同的工作, 每人承担一项工作, 则不同的选派方案有\\blank{50}种(用数值作答).", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "填空题", "ans": "", @@ -117076,7 +117391,9 @@ "content": "从$1,2,3,4,5,6,7,8,9$中任取$5$个不同的数, 中位数为$4$的取法有\\blank{50}种(用数值表示).", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合", + "加法原理与乘法原理" ], "genre": "填空题", "ans": "$30$", @@ -117249,7 +117566,8 @@ "content": "某班有$20$名女生和$19$名男生, 从中选出$5$人组成一个垃圾分类宣传小组, 要求女生和男生均不少于$2$人的选法共有\\bracket{20}.\n\\twoch{$\\mathrm{C}_{20}^2\\cdot \\mathrm{C}_{19}^2\\cdot \\mathrm{C}_{35}^1$}{$\\mathrm{C}_{39}^5-\\mathrm{C}_{20}^5-\\mathrm{C}_{19}^5$}{$\\mathrm{C}_{39}^5-\\mathrm{C}_{20}^1\\mathrm{C}_{19}^4-\\mathrm{C}_{20}^4\\mathrm{C}_{19}^{1}$}{$\\mathrm{C}_{20}^2\\mathrm{C}_{19}^3+\\mathrm{C}_{20}^3\\mathrm{C}_{19}^2$}", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "选择题", "ans": "", @@ -117736,7 +118054,8 @@ "content": "某班级要从$5$名男生和$3$名女生中选出$3$人参加公益活动, 则在选出的$3$人中男、女生均有的概率为\\blank{50}(结果用最简分数表示).", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "填空题", "ans": "", @@ -119495,7 +119814,8 @@ "K0803002B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "", @@ -120193,7 +120513,8 @@ "content": "首届中国国际进出口博览会在上海举行, 某高校拟派$4$人参加连续$5$天的志愿者活动, 其中甲连续参加$2$天, 其余每人各参加$1$天. 共有\\blank{50}种不同的安排方法(结果用数值表示).", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "填空题", "ans": "", @@ -120494,7 +120815,8 @@ "content": "某地区气象台统计, 该地区下雨的概率是$\\dfrac 4{15}$, 刮风的概率是$\\dfrac 25$, 既刮风又下雨的概率为$\\dfrac 1{10}$, 设事件$A$表示``该地区下雨'', 事件$B$表示``该地区刮风'', 那么$P(B|A)$等于\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "$\\dfrac 38$", @@ -120520,7 +120842,8 @@ "content": "已知盒中装有$3$只螺口灯泡与$7$只卡口灯泡, 这些灯泡的外形都相同且灯口向下放着, 现需要安装一只卡口灯泡, 电工师傅每次从盒中任取一只并且不放回, 则在他第$1$次抽到的是螺口灯泡的条件下, 第$2$次抽到的是卡口灯泡的概率为\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "$\\dfrac 79$", @@ -120546,7 +120869,8 @@ "content": "近年来, 新能源汽车技术不断推陈出新, 新产品不断涌现, 在汽车市场上影响力不断增大. 动力蓄电池技术作为新能源汽车的核心技术, 它的不断成熟也是推动新能源汽车发展的主要动力. 假定现在市售的某款新能源汽车上, 车载动力蓄电池充放电循环次数达到$2000$次的概率为$85\\%$, 充放电循环次数达到$2500$次的概率为$35\\%$. 若某用户的自用新能源汽车已经经过了$2000$次充电, 那么他的车能够充电$2500$次的概率为\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "$\\dfrac 7{17}$", @@ -120572,7 +120896,8 @@ "content": "将三颗骰子各掷一次, 记事件$A$为``三个点数都不相同'', $B$为``至少出现一个$6$点'', 则条件概率$P(A|B)$=\\blank{50}, $P(B|A)$=\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "$\\dfrac{60}{91}$, $\\dfrac 12$", @@ -120598,7 +120923,8 @@ "content": "袋中有大小完全相同的$2$个白球和$3$个黄球, 逐个不放回地摸出$2$个球, 设``第一次摸到白球''为事件$A$, ``摸到的$2$个球同色''为事件$B$, 则$P(B|A)$=\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "$\\dfrac 14$", @@ -120624,7 +120950,8 @@ "content": "已知$P(A)>0$, $P(B)>0$, $P(B|A)=P(B)$, 证明: $P(A|B)=P(A)$.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "略", @@ -120650,7 +120977,8 @@ "content": "*甲、乙、丙三人互相作传球训练, 第$1$次由甲将球传出, 每次传球时, 传球者都等可能地将球传给另外两个人中的任何一个, 求$4$次传球后球在甲手中的概率.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "$\\dfrac 38$", @@ -120676,7 +121004,8 @@ "content": "现在有$12$道四选一的单选题, 学生张三对其中$9$道题有思路, $3$道题完全没有思路. 有思路的题做对的概率为$0.9$, 没有思路的题只好任意猜一个答案, 猜对的概率为$0.25$, 张三从这$12$道题中随机选择$1$题, 则他做对该题的概率是\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "$\\dfrac{59}{80}$", @@ -120703,7 +121032,8 @@ "content": "两批同种规格的产品, 第一批占$40\\%$, 次品率为$5\\%$;第二批占$60\\%$, 次品率为$4\\%$, 将这两批产品混合, 从混合的产品中任取一件. 则这件产品时合格品的概率是\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "$0.956$", @@ -120730,7 +121060,8 @@ "content": "甲和乙两个箱子中各装有$10$个球, 其中甲箱中有$5$个红球、$5$个白球, 乙箱中有$8$个红球、$2$个白球. 掷一枚质地均匀的骰子, 如果点数为$1$或$2$, 从甲箱子随机摸出$1$个球; 如果点数为$3, 4, 5, 6$, 从乙箱子中随机摸出$1$个球, 则摸到红球的概率是\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "$0.7$", @@ -120757,7 +121088,8 @@ "content": "在$A$、$B$、$C$三个地区暴发了流感, 这三个地区分别有$6\\%$, $5\\%$, $4\\%$的人患了流感, 假设这三个地区的人口数的比为$5: 7: 8$, 现从这三个地区中任意选取一个人. 则这个人患流感的概率是\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "$0.0485$", @@ -120784,7 +121116,8 @@ "content": "甲、乙两人独立地向同一目标各射击一次, 已知甲命中目标的概率为$0.6$, 乙命中目标的概率为$0.5$, 则目标至少被命中一次时, 甲命中目标的概率是\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "$0.75$", @@ -120811,7 +121144,8 @@ "content": "设$P(A)>0$, 且$B$和$\\overline B$是对立事件, 求证: $P(\\overline B|A)=1-P(B|A)$.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "略", @@ -120838,7 +121172,8 @@ "content": "一批产品共有$100$件, 其中$5$件为不合格品, 收货方从中不放回地随机抽取产品进行检验, 并按以下规则判断是否接受这批产品; 如果抽检的第$1$件产品不合格, 则拒绝整批产品; 如果抽检的第一件产品合格, 则再抽$1$件, 如果抽检的第$2$件产品合格, 则接受整批产品, 否则拒绝整批产品, 求这批产品被拒绝的概率.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "$\\dfrac{97}{990}$", @@ -120865,7 +121200,8 @@ "content": "在孟德尔豌豆试验中, 子二代(数量充分大)的基因型为DD, Dd, dd, 其中D为显性基因, d为隐性基因, 且这三种基因型的比为$1: 2: 1$. 如果在子二代中任意选取$2$颗豌豆作为父代进行杂交试验, 那么第三代中基因型为dd的概率有多大?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "$\\dfrac 14$", @@ -120892,7 +121228,8 @@ "content": "长时间玩手机可能影响视力, 据调查, 某校学生大约$40\\%$的人近视, 而该校大约有$20\\%$的学生每天玩手机超过$1\\text{h}$, 这些人的近视率为$50\\%$. 现从每天玩手机不超过$1\\text{h}$的学生中任意调查一名学生, 求他的近视概率.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "$\\dfrac 38$", @@ -120919,7 +121256,8 @@ "content": "设随机变量$X$的概率分布列如下, 则$P(|X-2|=1)=$\\blank{50}.\n\\begin{center}\n $\\begin{pmatrix}\n 1 & 2 & 3 & 4\\\\ \n \\dfrac 16 & \\dfrac 14 & m & \\dfrac 13 \n \\end{pmatrix}$\n\\end{center}", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "$\\dfrac 5{12}$", @@ -120946,7 +121284,8 @@ "content": "已知离散型随机变量$X$的分布列为\n\\begin{center}\n $\\begin{pmatrix}\n 0 & 1 & 2 \\\\ \n 0.5 & 1-2q & q^2 \n \\end{pmatrix}$\n\\end{center}\n则常数$q=$\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "$1-\\dfrac{\\sqrt{2}}{2}$", @@ -120973,7 +121312,8 @@ "content": "一盒中有$12$个乒乓球, 其中$9$个新的, $3$个旧的, 从盒子中一次性任取$3$个球来用, 用完即为旧的, 用完后装回盒中, 此时盒中旧球个数$X$是一个随机变量, 则$P(X=4)$的值为\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "$\\dfrac{27}{220}$", @@ -121000,7 +121340,8 @@ "content": "离散型随机变量$X$的概率分布规律为$P(X=n)=\\dfrac{a}{n(n+1)}$($n=1, 2, 3, 4$), 其中$a$是常数, 则$P(\\dfrac 12''连接);\\\\\n(2) 在上面的句子中随机取一个单词, 用$X$表示取到的单词所包含的字母个数, 写出$X$的分布列;\\\\\n(3) 从上述单词中任选$2$个单词, 求其字母个数之和为$6$的概率.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "(1) $f(\\text{e})=\\dfrac{5}{29}$, $f(\\text{i})=\\dfrac{3}{29}$, $f(\\text{t})=\\dfrac{4}{29}$, $f(\\text{a})=\\dfrac{2}{29}$, $f(\\text{e})>f(\\text{t})>f(\\text{i})>f(\\text{a})$; (2) $\\begin{pmatrix} 2 & 3 & 4 & 5 \\\\ \\dfrac 29 & \\dfrac 49 & \\dfrac 29 & \\dfrac 19 \\end{pmatrix}$; (3) $\\dfrac 5{18}$.", @@ -121135,7 +121480,8 @@ "content": "已知$X$的分布列为\n\\begin{center}\n $\\begin{pmatrix}\n -1 & 0 & 1 \\\\ \n \\dfrac 12 & \\dfrac 13 & \\dfrac 16 \n \\end{pmatrix}$\n\\end{center}\n两个随机变量$X$, $Y$满足$X+2Y=4$, 则$E[X]=$\\blank{50}, $E[Y]=$\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "", @@ -121161,7 +121507,8 @@ "content": "``过大年, 吃水饺''是我国不少地方过春节的一大习俗. $2021$年春节前夕, $A$市某质量检测部门随机抽取了$100$包某种品牌的速冻水饺, 检测其某项质量指标值, 所得频率分布直方图如图.\n\\begin{center}\n \\begin{tikzpicture}[>=latex]\n \\draw [->] (0,0) -- (7,0) node [below] {质量指标值};\n \\draw [->] (0,0) -- (0,4) node [left] {$\\dfrac{\\text{频率}}{\\text{组距}}$};\n \\draw (0,0) node [below left] {$O$};\n \\draw (0,1) node [left] {$0.010$} -- (1,1) -- (1,0) node [below] {$10$};\n \\draw (1,1) -- (1,2) -- (2,2) -- (2,0) node [below] {$20$};\n \\draw (2,2) -- (2,3) -- (3,3) -- (3,0) node [below] {$30$};\n \\draw (3,2.5) -- (4,2.5) -- (4,0) node [below] {$40$};\n \\draw (4,1.5) -- (5,1.5) -- (5,0) node [below] {$50$};\n \\draw [dashed] (0,1.5) node [left] {$0.015$} -- (4,1.5);\n \\draw [dashed] (0,2) node [left] {$0.020$} -- (1,2);\n \\draw [dashed] (0,2.5) node [left] {$0.025$} -- (3,2.5);\n \\draw [dashed] (0,3) node [left] {$0.030$} -- (2,3);\n \\end{tikzpicture}\n\\end{center}\n(1) 求所抽取的$100$包速冻水饺该项质量指标值的样本平均数$\\overline x$(同一组中的数据用该组区间的中点值作代表);\\\\\n(2) 将频率视为概率, 若某人从该市某超市购买了$4$包这种品牌的速冻水饺, 记这$4$包速冻水饺中该项质量指标值位于$(10,30]$内的包数为$X$, 求$X$的分布列和期望.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -121185,7 +121532,8 @@ "content": "近年来, 祖国各地依托本地自然资源, 打造旅游产业, 旅游业正蓬勃发展. 景区与游客都应树立尊重自然、顺应自然、保护自然的生态文明理念, 合力使旅游市场走上规范有序且可持续的发展轨道. 某景区有一个自愿消费的项目: 在参观某特色景点入口处会为每位游客拍一张与景点的合影, 参观后, 在景点出口处会将刚拍下的照片打印出来, 游客可自由选择是否带走照片, 若带走照片则需支付$20$元, 没有被带走的照片会收集起来统一销毁. 该项目运营一段时间后, 统计出平均只有$30\\%$游客会选择带走照片. 为改善运营状况, 该项目组就照片收费与游客消费意愿关系做了市场调研, 发现收费与消费意愿有较强的线性相关性, 并统计出在原有的基础上, 价格每下调$1$元, 游客选择带走照片的可能性平均增加$0.05$. 假设平均每天约有$5000$人参观该特色景点, 每张照片的综合成本为$5$元, 假设每位游客是否购买照片相互独立.\\\\\n(1) 若调整为支付$10$元就可带走照片, 该项目每天的平均利润比调整前多还是少?\\\\\n(2) 要使每天的平均利润达到最大值, 应如何定价?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -121209,7 +121557,8 @@ "content": "某种大型医疗检查机器生产商, 对一次性购买$2$台机器的客户, 推出$2$种超过质保期后$2$年内的延保维修优惠方案.\\\\\n方案一: 交纳延保金$7000$元, 在延保的$2$年内可免费维修$2$次, 超过$2$次每次收取维修费$2000$元;\\\\\n方案二: 交纳延保金$10000$元, 在延保的$2$年内可免费维修$4$次, 超过$4$次每次收取维修费$1000$元.\\\\\n某医院准备一次性购买$2$台这种机器. 现需决策在购买机器时应购买哪种延保方案, 为此搜集并整理了$50$台这种机器超过质保期后延保$2$年内维修的次数, 得下表:\n\\begin{center}\n \\begin{tabular}{|c|c|c|c|c|}\n \\hline\n 维修次数 & $0$ & $1$ & $2$ & $3$\\\\ \\hline\n 台数 & $5$ & $10$ & $20$ & $15$\\\\ \\hline\n \\end{tabular}\n\\end{center}\n以这$50$台机器维修次数的频率代替$1$台机器维修次数发生的概率. 记$X$表示这$2$台机器超过质保期后延保的$2$年内共需维修的次数.\\\\\n(1) 求$X$的分布列;\\\\\n(2) 以方案一与方案二所需费用(所需延保金及维修费用之和)的期望值为决策依据, 医院选择哪种延保方案更合算?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -121233,7 +121582,8 @@ "content": "已知$X$的分布列为\n\\begin{center}\n $\\begin{pmatrix}\n -1 & 0 & 1 \\\\\n \\dfrac 12 & \\dfrac 13 & \\dfrac 16 \n \\end{pmatrix}$\n\\end{center}\n两个随机变量$X$, $Y$满足$X+2Y=4$, 则$D[X]=$\\blank{50}, $D[Y]=$\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "", @@ -121258,7 +121608,8 @@ "content": "五个自然数$1, 2, 3, 4, 5$按照一定的顺序排成一排.\\\\\n(1) 求$2$和$4$不相邻的概率;\\\\\n(2) 定义: 若两个数的和为$6$且相邻, 称这两个数为一组``友好数''. 随机变量X表示上述五个自然数组成的一个排列中``友好数''的组数, 求$X$的分布列、数学期望$E[X]$和方差$D[X]$.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -121281,7 +121632,8 @@ "content": "为推广滑雪运动, 某滑雪场开展滑雪促销活动. 该滑雪场的收费标准是: 滑雪时间不超过$1$小时免费, 超过$1$小时的部分每小时收费标准为$40$元(不足$1$小时的部分按$1$小时计算). 有甲、乙两人相互独立地来该滑雪场运动, 设甲、乙不超过$1$小时离开的概率分别为$\\dfrac 14$, $\\dfrac 16$; $1$小时以上且不超过$2$小时离开的概率分别为$\\dfrac 12$, $\\dfrac 23$; 两人滑雪时间都不会超过$3$小时.\\\\\n(1) 求甲、乙两人所付滑雪费用相同的概率;\\\\\n(2) 设甲、乙两人所付的滑雪费用之和为随机变量$X$(单位: 元), 求$X$的分布列与数学期望$E[X]$, 方差$D[X]$.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -121304,7 +121656,8 @@ "content": "甲、乙两人各射击$1$次, 击中目标的概率分别是$\\dfrac 23$和$\\dfrac 12$, 假设两人击中目标与否相互之间没有影响, 每人各次击中目标与否相互之间也没有影响, 若两人各射击$4$次, 则甲恰好有$2$次击中目标且乙恰好有$3$次击中目标的概率为\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "", @@ -121327,7 +121680,8 @@ "content": "在一次招聘中, 主考官要求应聘者从$18$道备选题中一次性随机抽取$9$道题, 并独立完成所抽取的$9$道题. 甲能正确完成每道题的概率为$\\dfrac 23$, 且每道题完成与否互不影响. 记甲能答对的题数为$X$, 则$X$的期望为\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "", @@ -121350,7 +121704,8 @@ "content": "设$X$为随机变量, 且$X\\sim B(n,p)$, 若随机变量$X$的数学期望$E[X]=4$, $D[X]=\\dfrac 43$, 则$P(X=2)=$\\blank{50}(结果用分数表示).", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "", @@ -121373,7 +121728,8 @@ "content": "某地区为贯彻习近平总书记关于``绿水青山就是金山银山''的理念, 鼓励农户利用荒坡种植果树. 某农户考察三种不同的果树苗$A,B,C$, 经引种试验后发现, 引种树苗$A$的自然成活率为$0.8$, 引种树苗$B,C$的自然成活率均为$p$($0.7\\le p\\le 0.9$).\\\\\n(1) 任取树苗$A,B,C$各一棵, 估计自然成活的棵数为$X$, 求$X$的分布列及数学期望$E[X]$;\\\\\n(2) 将(1)中的$E[X]$取得最大值时$p$的值作为$B$种树苗自然成活的概率. 该农户决定引种$n$棵$B$种树苗, 引种后没有自然成活的树苗中有$75\\%$的树苗可经过人工栽培技术处理, 处理后成活的概率为$0.8$, 其余的树苗不能成活.\\\\\n\\textcircled{1} 求一棵$B$种树苗最终成活的概率;\\\\\n\\textcircled{2} 若每棵树苗最终成活后可获利$300$元, 不成活的每棵亏损$50$元, 该农户为了获利不低于$20$万元, 问至少引种$B$种树苗多少棵?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -121396,7 +121752,8 @@ "content": "一款小游戏的规则如下: 每轮游戏要进行三次, 每次游戏都需要从装有大小相同的$2$个红球、$3$个白球的袋中随机摸出$2$个球, 若``摸出的两个球都是红球''出现$3$次获得$200$分, 若``摸出的两个球都是红球''出现$1$次或$2$次获得$20$分, 若``摸出的两个球都是红球''出现$0$次, 则扣除$10$分(即获得负$10$分).\\\\\n(1) 设每轮游戏中出现``摸出的两个球都是红球''的次数为$X$, 求$X$的分布列;\\\\\n(2) 许多玩过这款游戏的人发现, 若干轮游戏后, 与最初的分数相比, 分数没有增加, 反而减少了, 请运用概率统计的相关知识解释上述现象.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -121419,7 +121776,8 @@ "content": "一年之计在于春, 一日之计在于晨, 春天是播种的季节, 是希望的开端. 某种植户对一块地的$n$($n\\in \\mathbf{N}$, $n>0$)个坑进行播种, 每个坑播$3$粒种子, 每粒种子发芽的概率均为$\\dfrac 12$, 且每粒种子是否发芽相互独立. 对每一个坑而言, 如果至少有$2$粒种子发芽, 则不需要进行补播种, 否则要补播种.\\\\\n(1) 设恰有$3$个坑需要补种的概率为$f(n)$($n\\in \\mathbf{N}$, $n\\ge 3$), 当$n$取何值时, $f(n)$取得最大值? $f(n)$的最大值为多少?\\\\\n(2) 当$n=4$时, 用$X$表示要补播种的坑的个数, 求$X$的分布列与数学期望.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -121442,7 +121800,8 @@ "content": "$2019$年$3$月$5$日, 国务院总理李克强作的政府工作报告中, 提到要``惩戒学术不端, 力戒浮躁之风''. 教育部$2014$年印发的《博士硕士学位论文抽检办法》通知中规定: 每篇抽检的学位论文送$3$位同行专家进行评议, $3$位专家中有$2$位以上(含$2$位)专家评议意见为``不合格''的学位论文, 将认定为``存在问题学位论文'', 有且仅有$1$位专家评议意见为``不合格''的学位论文, 将再送另外$2$位同行专家(不同于前$3$位专家)进行复评, $2$位复评专家中有$1$位以上(含$1$位)专家评议意见为``不合格''的学位论文, 将认定为``存在问题学位论文''. 设每篇学位论文被每位专家评议为``不合格''的概率均为$p$($00$)次. 在抽样结束时, 若已抽到的黄色汽车数以$X$表示, 求$X$的分布列和数学期望.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -121484,7 +121844,8 @@ "content": "河南省三门峡市成功入围``十佳魅力中国城市'', 吸引了大批投资商的目光, 一些投资商积极准备投入到``魅力城市''的建设之中. 某投资公司准备在$2022$年年初将$400$万元投资到三门峡下列两个项目中的一个之中.\\\\\n项目一: 天坑院是黄土高原地域独具特色的民居形式, 是人类``穴居''发展史演变的实物见证. 现准备投资建设$20$个天坑院, 每个天坑院投资$20$万元, 假设每个天坑院是否盈利是相互独立的, 据市场调研, 到$2024$年底每个天坑院盈利的概率为$p$($0=latex]\n \\draw [->] (-1,0) -- (0,0) node [below left] {$A$};\n \\draw [->] (4,0) node [below right] {$B$}-- (5,0); \n \\draw (0,0) -- (0,-1.5) -- (0.5,-1.5) (1.5,-1.5) -- (2.5,-1.5) (3.5,-1.5) -- (4,-1.5) -- (4,0);\n \\draw (0,0) -- (0,1.5) -- (1.5,1.5) (2.5,1.5) -- (4,1.5) -- (4,0);\n \\draw (0,0) -- (1.75,0) (2.25,0) -- (4,0);\n \\draw (1.75,0) --++ (30:0.55);\n \\filldraw [fill = white, draw = black] (1.75,0) circle (0.05);\n \\draw (0.5,-1.5) --++ (0,0.3) --++ (0.25,0) ++ (0.5,0) --++ (0.25,0) --++ (0,-0.3) (0.5,-1.5) --++ (0,-0.3) --++ (0.25,0) ++ (0.5,0) --++ (0.25,0) --++ (0,0.3);\n \\draw (2.5,-1.5) --++ (0,0.3) --++ (0.25,0) ++ (0.5,0) --++ (0.25,0) --++ (0,-0.3) (2.5,-1.5) --++ (0,-0.3) --++ (0.25,0) ++ (0.5,0) --++ (0.25,0) --++ (0,0.3);\n \\draw (1.5,1.5) --++ (0,0.5) --++ (0.25,0) ++ (0.5,0) --++ (0.25,0) --++ (0,-0.5) (1.5,1.5) --++ (0,-0.5) --++ (0.25,0) ++ (0.5,0) --++ (0.25,0) --++ (0,0.5) (1.5,1.5) --++ (0.25,0) ++ (0.5,0) --++ (0.25,0);\n \\draw (0.75,-1.2) --++ (30:0.55);\n \\filldraw [fill = white, draw = black] (0.75,-1.2) circle (0.05);\n \\draw (0.75,-1.8) --++ (30:0.55);\n \\filldraw [fill = white, draw = black] (0.75,-1.8) circle (0.05);\n \\draw (2.75,-1.2) --++ (30:0.55);\n \\filldraw [fill = white, draw = black] (2.75,-1.2) circle (0.05);\n \\draw (2.75,-1.8) --++ (30:0.55);\n \\filldraw [fill = white, draw = black] (2.75,-1.8) circle (0.05);\n \\draw (1.75,1) --++ (30:0.55);\n \\filldraw [fill = white, draw = black] (1.75,1) circle (0.05);\n \\draw (1.75,1.5) --++ (30:0.55);\n \\filldraw [fill = white, draw = black] (1.75,1.5) circle (0.05);\n \\draw (1.75,2) --++ (30:0.55);\n \\filldraw [fill = white, draw = black] (1.75,2) circle (0.05);\n \\filldraw (0,0) circle (0.05) (4,0) circle (0.05) (1.5,1.5) circle (0.05) (2.5,1.5) circle (0.05) (0.5,-1.5) circle (0.05) (1.5,-1.5) circle (0.05) (2.5,-1.5) circle (0.05) (3.5,-1.5) circle (0.05);\n \\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第八单元" + "第八单元", + "加法原理与乘法原理" ], "genre": "填空题", "ans": "", @@ -185747,7 +186144,8 @@ "content": "若集合$M=\\{-1,1,2\\}$, 且$a,b,r\\in M$, 则$(x-a)^2+(y-b)^2=r^2$所表示的不同圆共有\\blank{50}个.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "加法原理与乘法原理" ], "genre": "填空题", "ans": "", @@ -185768,7 +186166,8 @@ "content": "若$a\\in \\{-1,2,3\\}$, $b\\in \\{0,3,4,5\\}$, $R\\in \\{1,2\\}$, 则方程$(x-a)^2+(y-b)^2=R^2$所表示的不同圆有\\blank{50}个.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "加法原理与乘法原理" ], "genre": "填空题", "ans": "", @@ -185789,7 +186188,8 @@ "content": "某乒乓球队行男运动员$7$人, 女运动员$6$人, 从中选出一名担任队长, 共有\\blank{50}种不同方案; 从中派出$2$人参加男女混合双打, 共有\\blank{50}种不同方案.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "加法原理与乘法原理" ], "genre": "填空题", "ans": "", @@ -185810,7 +186210,8 @@ "content": "若$m\\in \\{-2,-1,0,1,2,3\\}$, $n\\in \\{-3,-2,-1,0,1,2\\}$, 且方程$\\dfrac{x^2}m+\\dfrac{y^2}n=1$是表示中心在原点的双曲线, 则表示不同的双曲线最多有\\blank{50}条.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "加法原理与乘法原理" ], "genre": "填空题", "ans": "", @@ -185831,7 +186232,8 @@ "content": "$3$张卡片的正反面分别写有数字$1$和$2$, $3$和$4$, $5$和$6$, 若将$3$张卡片并列, 可得到\\blank{50}个不同的三位数($6$不能作$9$用).", "objs": [], "tags": [ - "第八单元" + "第八单元", + "加法原理与乘法原理" ], "genre": "填空题", "ans": "", @@ -185852,7 +186254,8 @@ "content": "从$2, 3, 5, 7$这$4$个数字中, 任取两个分别作为分数的分子与分母.\\\\\n(1) 能得到几个不同的分数?\\\\\n(2) 其中有几个是真分数? 几个是假分数?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "加法原理与乘法原理" ], "genre": "解答题", "ans": "", @@ -185876,7 +186279,8 @@ "K0618003B" ], "tags": [ - "第八单元" + "第八单元", + "加法原理与乘法原理" ], "genre": "选择题", "ans": "B", @@ -185902,7 +186306,8 @@ "content": "有一排$5$个信号的显示窗, 每个窗可亮红灯、绿灯或不亮灯, 则共可发出的不同信号有\\bracket{20}.\n\\fourch{$2^5$种}{$5^2$种}{$3^5$种}{$5^3$种}", "objs": [], "tags": [ - "第八单元" + "第八单元", + "加法原理与乘法原理" ], "genre": "选择题", "ans": "", @@ -185923,7 +186328,8 @@ "content": "$4$位学生各写一张贺卡, 放在一起, 然后每人从中各取一张, 但不能取自己写的那一张贺卡, 则不同的取法共有\\bracket{20}.\n\\fourch{$9$种}{$12$种}{$16$种}{$24$种}", "objs": [], "tags": [ - "第八单元" + "第八单元", + "加法原理与乘法原理" ], "genre": "选择题", "ans": "", @@ -185944,7 +186350,8 @@ "content": "$3$封不同的信, 投入$4$个信箱, 则并有不同的投法\\blank{50}种.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "加法原理与乘法原理" ], "genre": "填空题", "ans": "", @@ -185965,7 +186372,8 @@ "content": "$4$个学生报名参加跳高, 跳远, 游泳比赛, 每人限报$1$项, 则不同的报名方法共有\\blank{50}种.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "加法原理与乘法原理" ], "genre": "填空题", "ans": "", @@ -185986,7 +186394,8 @@ "content": "若集合$A=\\{a_1,a_2,a_3,a_4,a_5\\}$, $B=\\{b_1,b_2,b_3\\}$, 则从集合$A$到$B$可建立\\blank{50}个不同的映射, 从集合$B$到集合$A$可建立\\blank{50}个不同的映射.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "加法原理与乘法原理" ], "genre": "填空题", "ans": "", @@ -186007,7 +186416,8 @@ "content": "如图, 用4种不同的颜色涂入图中编兮为1, 2, 3, 4的正方形, 要求每个正方形只涂一种颜色, 且有公共边的两个正方形颜色不同, 则共有多少种不同的涂法?\n\\begin{center}\n \\begin{tikzpicture}\n \\draw (0,0) rectangle (2,2);\n \\draw (1,0) -- (1,2) (0,1) -- (2,1);\n \\draw (0.5,1.5) node {$1$} (1.5,1.5) node {$2$} (0.5,0.5) node {$3$} (1.5,0.5) node {$4$};\n \\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第八单元" + "第八单元", + "加法原理与乘法原理" ], "genre": "解答题", "ans": "", @@ -186028,7 +186438,8 @@ "content": "从$1$到$100$的自然数中, 每次取两个不同的数相加, 使它们的和不大于$100$, 有几种取法? ($3+6$与$4+5$算作不同的取法).", "objs": [], "tags": [ - "第八单元" + "第八单元", + "加法原理与乘法原理" ], "genre": "解答题", "ans": "", @@ -186049,7 +186460,8 @@ "content": "从$1$到$200$这$200$个自然数中, 各个数位上都不含有数字$8$的数有几个?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "加法原理与乘法原理" ], "genre": "解答题", "ans": "", @@ -186070,7 +186482,8 @@ "content": "有一角硬币$3$枚, 贰元币$6$张, 百元币$4$张, 共可组成多少种不同的币值.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "加法原理与乘法原理" ], "genre": "解答题", "ans": "", @@ -186091,7 +186504,8 @@ "content": "设$a\\in \\mathbf{N}$, 且$a<27$, 则$(27-a)(28-a)\\cdots (34-a)$等于\\bracket{20}.\n\\fourch{$\\mathrm{P}_{27-a}^8$}{$\\mathrm{P}_{34-a}^{27-a}$}{$\\mathrm{P}_{34-a}^7$}{$\\mathrm{P}_{34-a}^8$}", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "选择题", "ans": "", @@ -186112,7 +186526,8 @@ "content": "$6$人站成一排照相, 其中甲、乙、丙三人要站在一起, 且要求乙、丙分别站在甲的两边, 则不同的排法种数为\\bracket{20}.\n\\fourch{12}{24}{48}{144}", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "选择题", "ans": "", @@ -186133,7 +186548,8 @@ "content": "记$8$个同学排成一排的排列数为$m$, $8$个同学排成前后两排(前排$3$人, 后排$5$人)的排列数为$n$, 则$m$, $n$的大小关系是\\bracket{20}.\n\\fourch{$m=n$}{$m>n$}{$m\\mathrm{C}_{n-2}^3+2\\mathrm{C}_{n-2}^2+n-2$.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "解答题", "ans": "", @@ -188107,7 +188622,8 @@ "content": "解不等式: $\\mathrm{C}_{21}^{x-4}<\\mathrm{C}_{21}^{x-2}<\\mathrm{C}_{21}^{x-1}$.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "解答题", "ans": "", @@ -188130,7 +188646,8 @@ "content": "解不等式: $\\mathrm{C}_k^0+\\mathrm{C}_k^1+2\\mathrm{C}_k^2+3\\mathrm{C}_k^3+\\cdots +k\\mathrm{C}_k^k<500$.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "解答题", "ans": "", @@ -188151,7 +188668,8 @@ "content": "解方程: $\\mathrm{C}_{16}^{x^2-x}=\\mathrm{C}_{16}^{5x-5}$.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "解答题", "ans": "", @@ -188172,7 +188690,8 @@ "content": "解方程: $\\mathrm{C}_{x+3}^{x+1}=\\mathrm{C}_{x+1}^{x-1}+\\mathrm{C}_{x+1}^x+\\mathrm{C}_x^{x-2}$.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "解答题", "ans": "", @@ -188193,7 +188712,8 @@ "content": "计算: $\\mathrm{C}_{2n}^{17-n}+\\mathrm{C}_{13+n}^{3n}$.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "解答题", "ans": "", @@ -188216,7 +188736,8 @@ "content": "计算: $\\mathrm{C}_{3n}^{38-n}+\\mathrm{C}_{21+n}^{3n}$.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "解答题", "ans": "", @@ -188239,7 +188760,8 @@ "content": "化简: $1\\cdot 1!+2\\cdot 2!+\\cdots +10\\cdot 10!$.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "解答题", "ans": "", @@ -188260,7 +188782,8 @@ "content": "求证: $\\dfrac 1{k!}-\\dfrac 1{(k+1)!}=\\dfrac k{(k+1)!}$.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "解答题", "ans": "", @@ -188281,7 +188804,8 @@ "content": "化简: $\\dfrac 1{2!}+\\dfrac 2{3!}+\\cdots +\\dfrac n{(n+1)!}$.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "解答题", "ans": "", @@ -188302,7 +188826,8 @@ "content": "求证: $\\dfrac{k+2}{k!+(k+1)!+(k+2)!}=\\dfrac 1{(k+1)!}-\\dfrac 1{(k+2)!}$.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "解答题", "ans": "", @@ -188323,7 +188848,8 @@ "content": "求和: ; $\\dfrac 3{1!+2!+3!}+\\dfrac 4{2!+3!+4!}+\\cdots +\\dfrac{n+2}{n!+(n+1)!+(n+2)!}$.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "解答题", "ans": "", @@ -188346,7 +188872,8 @@ "content": "求证: $\\mathrm{C}_n^k=\\mathrm{C}_2^0\\mathrm{C}_{n-2}^k+\\mathrm{C}_2^1\\mathrm{C}_{n-2}^{k-1}+\\mathrm{C}_2^2\\mathrm{C}_{n-2}^{k-2}$($k\\ge 2$).", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "解答题", "ans": "", @@ -188367,7 +188894,8 @@ "content": "求证: $n!+\\dfrac{(n+1)!}{1!}+\\dfrac{(n+2)!}{2!}+\\cdots +\\dfrac{(n+m)!}{m!}=n!\\mathrm{C}_{n+m+1}^{n+1}$.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "解答题", "ans": "", @@ -188388,7 +188916,9 @@ "content": "$n$个不同的球放入$n$个不同的盒子中, 若恰好有一个盒子是空盒, 则共有几种不同的放法?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合", + "排列" ], "genre": "解答题", "ans": "", @@ -188409,7 +188939,8 @@ "content": "从集合$M=\\{1,2,3,4,5\\}$到集合$N=\\{a,b,c\\}$的映射, 要求集合$N$中的元素在集合$M$中都有原像, 这样的映射有几种?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "解答题", "ans": "", @@ -188430,7 +188961,8 @@ "content": "如图, $A,B,C\\in l_1$, $D,E,F,G\\in l_2$, $H$不属于$l_1\\cup l_2$, 以这$8$个点中的$3$个点为顶点, 最多可作多少个不同的三角形?\n\\begin{center}\n \\begin{tikzpicture}[scale = 0.6]\n \\draw (0,0) -- (5,0) (0,0) -- (45:5);\n \\filldraw (1,0) circle (0.05) node [below] {$D$} (2,0) circle (0.05) node [below] {$E$} (3,0) circle (0.05) node [below] {$F$} (4,0) circle (0.05) node [below] {$G$} (4.5,1.5) circle (0.05) node [right] {$H$};\n \\draw (45:1) circle (0.05) node [above] {$A$} (45:2) circle (0.05) node [above] {$B$} (45:3) circle (0.05) node [above] {$C$};\n \\draw (5,0) node [right] {$l_2$} (45:5) node [right] {$l_1$};\n \\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第八单元" + "第八单元", + "加法原理与乘法原理" ], "genre": "解答题", "ans": "", @@ -188451,7 +188983,9 @@ "content": "$\\angle AOB$的两边$OA$, $OB$上分别有异于顶点$O$的$5$个点和$6$个点, 这$12$个点(连同$O$点)可作几条不同直线和几个不同的三角形?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合", + "加法原理与乘法原理" ], "genre": "解答题", "ans": "", @@ -188472,7 +189006,9 @@ "content": "在$ABCD$中, $M$, $N$是边$AB$的三等分点, $P$是边$CD$的中点, 从$A$, $B$, $C$, $D$, $M$, $N$, $P$这$7$个点中选$3$个作为三角形的顶点, 一共可以构成几个不同的三角形? 其中面积最小的三角形有几个?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合", + "加法原理与乘法原理" ], "genre": "解答题", "ans": "", @@ -188496,7 +189032,8 @@ "K0618007B" ], "tags": [ - "第八单元" + "第八单元", + "加法原理与乘法原理" ], "genre": "填空题", "ans": "", @@ -188517,7 +189054,9 @@ "content": "正方体有$8$个顶点, 每$3$点确定$1$个平面, 一共可确定多少个平面?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合", + "加法原理与乘法原理" ], "genre": "解答题", "ans": "", @@ -188538,7 +189077,8 @@ "content": "从集合$\\{51,52,53,\\cdots ,99\\}$中任选$2$个数, 使这$2$个数的和为偶数, 有多少种不同的选法?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "解答题", "ans": "", @@ -188559,7 +189099,8 @@ "content": "从$1$到$100$的自然数中, 每次取两个不同的数相加, 使它们的和不大于$100$, 有几种不同的取法($1+4$与$2+3$算不同的取法, $2+3$与$3+2$算相同的取法)?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "加法原理与乘法原理" ], "genre": "解答题", "ans": "", @@ -188580,7 +189121,9 @@ "content": "从$1$到$18$这$18$个自然数中任选$3$个, 使它们的和是$3$的倍数, 有几种选法?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合", + "加法原理与乘法原理" ], "genre": "解答题", "ans": "", @@ -188601,7 +189144,8 @@ "content": "从$5$个男乒乓球运动员和$4$个女乒乓球运动员中选出$2$男、$2$女进行乒乓球混合双打, 有多少种不同的分组方法?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "解答题", "ans": "", @@ -188622,7 +189166,9 @@ "content": "有编号为$1, 2, 3, 4, 5, 6, 7$的$7$个球和编号为$1, 2, 3, 4, 5, 6, 7$的$7$只盒子, 将这$7$个球放入这$7$只盒子中, 要求每只盒子放$1$个, 恰使其中$4$个球的编号与盒子的编号相同, 一共有多少种不同的投放方法?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列", + "加法原理与乘法原理" ], "genre": "解答题", "ans": "", @@ -188643,7 +189189,8 @@ "content": "$9$件相同的奖品分给$3$个学生, 每人至少分得$2$件奖品, 一共存几种不同的分法?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "加法原理与乘法原理" ], "genre": "解答题", "ans": "", @@ -188664,7 +189211,8 @@ "content": "$7$个相同的球任意放入$4$个不同的盒子中, 每个盒子至少有$1$个球的不同放法有几种?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "加法原理与乘法原理" ], "genre": "解答题", "ans": "", @@ -188685,7 +189233,8 @@ "content": "在连续的$6$次射击中, 恰好命中$4$次的情形有多少种?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "解答题", "ans": "", @@ -188706,7 +189255,8 @@ "content": "在所有的三位数中(数字允许重复), 百位数字, 十位数字, 个位数字依次减小的有多少个? 仅是个位数字比百位数字小的有多少个?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "解答题", "ans": "", @@ -188727,7 +189277,8 @@ "content": "圆上有$10$个点, 每两点连成一条线段, 这些线段在圆内最多有多少个交点?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "解答题", "ans": "", @@ -188748,7 +189299,8 @@ "content": "将分别写有$a,b,c,d,e,1,2,3,4,5$的$10$张纸片排成一列, 要求$5$在最前, $1$在最后, 且数字从大到小, 字母按英文字母表的先后顺序排列, 则有多少种不同的排法?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "解答题", "ans": "", @@ -188769,7 +189321,8 @@ "content": "从$1, 2,\\cdots, 10$这$10$个数中任取$3$个互不相邻的自然数, 有儿种不同的取法?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "加法原理与乘法原理" ], "genre": "解答题", "ans": "", @@ -188790,7 +189343,8 @@ "content": "从$6$个运动员中, 选出$4$人参加$4\\times 100$米接力赛跑, 若其中甲、乙两人都不能跑第一棒, 共有多少种参赛方案?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "解答题", "ans": "", @@ -188813,7 +189367,8 @@ "content": "从$7$名运动员中, 选出$4$人参加$4\\times 100$米接力赛跑, 若要求甲、乙两人都不跑中间两棒, 共有多少种参赛方案?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "解答题", "ans": "", @@ -188836,7 +189391,8 @@ "content": "有$6$名运动员参加$4\\times 100$米接力跑, 其中甲不能跑第一棒, 乙不跑第四棒, 共有多少种参赛的方法?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "解答题", "ans": "", @@ -188857,7 +189413,8 @@ "content": "$3$天中, 考政治、语文、外语、数学、物理和化学$6$科.\\\\\n(1)每天考一文一理, 有几种不同的安排方法?\\\\\n(2)每天考一文一理, 且语文、数学不能同一天考, 有几种不同的安排方法?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "解答题", "ans": "", @@ -188878,7 +189435,9 @@ "content": "在无重复数字的四位数中, 其中恰有$2$个奇数数字和$2$个偶数数字的四位数共有多少个?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合", + "排列" ], "genre": "解答题", "ans": "", @@ -188899,7 +189458,9 @@ "content": "从$1, 3, 5, 7$这$4$个数字中任取$3$个, 从$0, 2, 4$这$3$个数字中任取$2$个, 共可组成多少个无重复数字的五位数?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合", + "排列" ], "genre": "解答题", "ans": "", @@ -188922,7 +189483,8 @@ "content": "$10$个人分乘$3$辆汽车, 要求甲车坐$5$人, 乙车坐$3$人, 丙车坐$2$人, 有多少种不同的乘车方法?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "解答题", "ans": "", @@ -188945,7 +189507,8 @@ "content": "某市今年有$8$项重点工程需要建设, 由甲、乙、丙、丁$4$个建筑公司承包, 若要求甲承包$3$项, 乙承包$1$项, 丙、丁各承包$2$项, 则共有多少种不同的承包方案?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "解答题", "ans": "", @@ -188966,7 +189529,9 @@ "content": "有$6$本不同的书, 分给甲、乙、丙$3$人, 按下列要求, 各有几种不同的分法:\\\\\n(1) 甲得$1$本, 乙得$2$本, 丙得$3$本;\\\\\n(2) 每人$2$本;\\\\\n(3) $1$人$1$本, $1$人$2$本, $1$人$3$本.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合", + "加法原理与乘法原理" ], "genre": "解答题", "ans": "", @@ -188987,7 +189552,8 @@ "content": "已知集合$A$和集合$B$各含有$12$个元素, $A\\cap B$含有$4$个元素, 试求同时满足下列两个条件的集合$C$的个数:\\\\\n(1) $C\\subset (A\\cup B)$, 且$C$中含有$3$个元素;\\\\\n(2) $C\\cap A\\ne \\varnothing$.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "解答题", "ans": "", @@ -189008,7 +189574,8 @@ "content": "有翻译$8$人, 其中$3$人只会英语, $2$人只会日语, 其余$3$人既会英语又会日语, 现从中选$6$人, 安排$3$人翻译英语, $3$人翻译日语, 则不同的安排方法有多少种?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "解答题", "ans": "", @@ -189193,7 +189760,8 @@ "content": "求证: $4^n-4^{n-1}\\mathrm{C}_n^1+4^{n-2}\\mathrm{C}_n^2-4^{n-3}\\mathrm{C}_n^3+\\cdots +4(-1)^{n-1}\\mathrm{C}_n^{n-1}+(-1)^n\\mathrm{C}_n^n=3^n$($n\\in \\mathbf{N}$).", "objs": [], "tags": [ - "第八单元" + "第八单元", + "二项式定理" ], "genre": "解答题", "ans": "", @@ -189214,7 +189782,8 @@ "content": "求证: $1-\\mathrm{C}_n^2+\\mathrm{C}_n^4-\\mathrm{C}_n^6+\\mathrm{C}_n^8-\\mathrm{C}_n^{10}+\\cdots =(\\sqrt 2)^n\\cos \\dfrac{n\\pi }4$,\n$\\mathrm{C}_n^1-\\mathrm{C}_n^3+\\mathrm{C}_n^5-\\mathrm{C}_n^7+\\mathrm{C}_n^9-\\mathrm{C}_n^{11}+\\cdots =(\\sqrt 2)^n\\sin \\dfrac{n\\pi }4$.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "二项式定理" ], "genre": "解答题", "ans": "", @@ -189235,7 +189804,8 @@ "content": "求证: $\\mathrm{C}_n^1+2\\mathrm{C}_n^2+3\\mathrm{C}_n^3+\\cdots +n\\mathrm{C}_n^n=n\\cdot 2^{n-1}$($n\\in \\mathbf{N}$).", "objs": [], "tags": [ - "第八单元" + "第八单元", + "二项式定理" ], "genre": "解答题", "ans": "", @@ -189256,7 +189826,8 @@ "content": "求证: $\\mathrm{C}_n^0+\\dfrac 12\\mathrm{C}_n^1+\\dfrac 13\\mathrm{C}_n^2+\\cdots +\\dfrac 1{n+1}\\mathrm{C}_n^n=\\dfrac 1{n+1}(2^{n+1}-1)$($n\\in \\mathbf{N}$).", "objs": [], "tags": [ - "第八单元" + "第八单元", + "二项式定理" ], "genre": "解答题", "ans": "", @@ -189277,7 +189848,8 @@ "content": "求证$\\mathrm{C}_n^0\\mathrm{C}_n^1+\\mathrm{C}_n^1\\mathrm{C}_n^2+\\cdots +\\mathrm{C}_n^{n-1}\\mathrm{C}_n^n=\\dfrac{(2n)!}{(n-1)!(n+1)!}$.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "二项式定理" ], "genre": "解答题", "ans": "", @@ -189300,7 +189872,8 @@ "content": "求证: $(\\mathrm{C}_n^0)^2+(\\mathrm{C}_n^1)^2+(\\mathrm{C}_n^2)^2+\\cdots +(\\mathrm{C}_n^n)^2=\\mathrm{C}_{2n}^n$($n\\in \\mathbf{N}$).", "objs": [], "tags": [ - "第八单元" + "第八单元", + "二项式定理" ], "genre": "解答题", "ans": "", @@ -189321,7 +189894,8 @@ "content": "求$53^{53}$除以$9$的余数.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "二项式定理" ], "genre": "填空题", "ans": "", @@ -189342,7 +189916,8 @@ "content": "求证: $n^{n-1}-1$能被$(n-1)^2$整除($n\\ge 3$, $n\\in \\mathbf{N}$).", "objs": [], "tags": [ - "第八单元" + "第八单元", + "二项式定理" ], "genre": "解答题", "ans": "", @@ -189363,7 +189938,8 @@ "content": "求证: $2<(1+\\dfrac 1n)^n<3$($n\\ge 2$, $n\\in \\mathbf{N}$).", "objs": [], "tags": [ - "第八单元" + "第八单元", + "二项式定理" ], "genre": "解答题", "ans": "", @@ -189560,7 +190136,8 @@ "content": "$(x+1)^4-4(x+1)^3+6(x+1)^2-4(x+1)+1$等于\\bracket{20}.\n\\fourch{$x^4$}{$-x^4$}{$1$}{$-1$}", "objs": [], "tags": [ - "第八单元" + "第八单元", + "二项式定理" ], "genre": "选择题", "ans": "", @@ -189919,7 +190496,8 @@ "content": "若$(4x-1)^6=a_6x^6+a_5x^5+a_4x^4+a_3x^3+a_2x^2+a_1x+a_0$, 则$a_6+a_5+a_4+a_3+a_2+a_1+a_0$的值等于\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "二项式定理" ], "genre": "填空题", "ans": "", @@ -189940,7 +190518,8 @@ "content": "若$(1-2x)^6=a_0+a_1x+a_2x^2+a_3x^3+a_4x^4+a_5x^5+a_6x^6$, 则$a_6-a_5+a_4-a_3+a_2-a_1$的值等于\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "二项式定理" ], "genre": "填空题", "ans": "", @@ -190009,7 +190588,8 @@ "content": "$1+7\\mathrm{C}_n^1+7^2\\mathrm{C}_n^2+7^3\\mathrm{C}_n^3+\\cdots+7^n\\mathrm{C}_n^n=$\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "二项式定理" ], "genre": "填空题", "ans": "", @@ -190030,7 +190610,8 @@ "content": "$1-2\\mathrm{C}_n^1+4\\mathrm{C}_n^2-\\cdots +(-2)^n\\mathrm{C}_n^n=$\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "二项式定理" ], "genre": "填空题", "ans": "", @@ -190051,7 +190632,8 @@ "content": "$3+3^{n-1}\\mathrm{C}_n^1+3^{n-2}\\mathrm{C}_n^2+\\cdots +3\\mathrm{C}_n^{n-1}+\\mathrm{C}_n^n=$\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "二项式定理" ], "genre": "填空题", "ans": "", @@ -190072,7 +190654,8 @@ "content": "$\\mathrm{C}_{21}^0-\\mathrm{C}_{21}^2+\\mathrm{C}_{21}^4-\\mathrm{C}_{21}^6+\\cdots +\\mathrm{C}_{21}^{16}-\\mathrm{C}_{21}^{18}+\\mathrm{C}_{21}^{20}=$\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "二项式定理" ], "genre": "填空题", "ans": "", @@ -190203,7 +190786,8 @@ "content": "若集合$P=\\{\\text{所有小于}1993\\text{的正奇数}\\}$, 则$P$的非空真子集的个数是\\bracket{20}.\n\\fourch{$2^{996}$}{$2^{996}-2$}{$2^{996}-1$}{$2^{995}$}", "objs": [], "tags": [ - "第八单元" + "第八单元", + "加法原理与乘法原理" ], "genre": "选择题", "ans": "", @@ -190290,7 +190874,8 @@ "content": "在$(x+1)(2x+1)(3x+1)\\cdots (nx+1)$的展开式中, $x$的一次项的系数是\\bracket{20}.\n\\fourch{$\\mathrm{C}_n^1$}{$\\mathrm{C}_n^2$}{$\\mathrm{C}_{n+1}^1$}{$\\mathrm{C}_{n+1}^2$}", "objs": [], "tags": [ - "第八单元" + "第八单元", + "二项式定理" ], "genre": "选择题", "ans": "", @@ -190333,7 +190918,8 @@ "content": "$55^{55}$被$8$除所得的余数是\\bracket{20}.\n\\fourch{$7$}{$-7$}{$1$}{$-1$}", "objs": [], "tags": [ - "第八单元" + "第八单元", + "二项式定理" ], "genre": "选择题", "ans": "", @@ -190879,7 +191465,8 @@ "content": "在$(x+1)(x+2)(x+3)\\cdots (x+10)$的展开式中, $7$的系数是多少? $x^8$的系数又是多少?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "二项式定理" ], "genre": "解答题", "ans": "", @@ -190900,7 +191487,8 @@ "content": "求$(x+1)(x+2)(x+3)\\cdots (x+n)$展开式中含$x^{n-2}$项的系数.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "二项式定理" ], "genre": "解答题", "ans": "", @@ -191075,7 +191663,8 @@ "content": "若$a$为常数, 则$\\displaystyle\\lim_{n\\to\\infty}\\dfrac{a+\\mathrm{C}_n^1+\\mathrm{C}_n^2+\\cdots +\\mathrm{C}_n^n}{2^n}$的值等于\\bracket{20}.\n\\fourch{$0$}{$\\dfrac 12$}{$1$}{$\\dfrac a2$}", "objs": [], "tags": [ - "第八单元" + "第八单元", + "二项式定理" ], "genre": "选择题", "ans": "", @@ -191118,7 +191707,8 @@ "content": "设$(1-2x)^8=a_0+a_1x+a_2x^2+\\cdots +a_8x^8$, 则$|a_0|+|a_1|+|a_2|+\\cdots +|a_8|$是\\bracket{20}.\n\\fourch{$-1$}{$1$}{$2^8$}{$3^8$}", "objs": [], "tags": [ - "第八单元" + "第八单元", + "二项式定理" ], "genre": "选择题", "ans": "", @@ -191163,7 +191753,8 @@ "content": "若$2000<\\mathrm{C}_n^1+\\mathrm{C}_n^2+\\mathrm{C}_n^3+\\cdots +\\mathrm{C}_n^n<3000$, 则$n=$\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "二项式定理" ], "genre": "填空题", "ans": "", @@ -191184,7 +191775,8 @@ "content": "若$x^4-3x^3+x^2+1=a(x+1)^4+b(x+1)^3+c(x+1)^2+d(x+1)+6$, 则$b=$\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "二项式定理" ], "genre": "填空题", "ans": "", @@ -191205,7 +191797,9 @@ "content": "设含有$10$个元素的集合的全部子集为$S$, 其中由$3$个元素组成的子集数为$T$, 则$\\dfrac TS$的值为\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合", + "加法原理与乘法原理" ], "genre": "填空题", "ans": "", @@ -191226,7 +191820,8 @@ "content": "设$(1+x)+(1+x)^2+(1+x)^3+\\cdots +(1+x)^n=b_0+b_1x+b_2x^2+\\cdots +b_nx^n$, 且$b_0+b_1+\\cdots +b_n=30$, 则自然数$n$的值等于\\bracket{20}.\n\\fourch{$4$}{$5$}{$6$}{$8$}", "objs": [], "tags": [ - "第八单元" + "第八单元", + "二项式定理" ], "genre": "选择题", "ans": "", @@ -191269,7 +191864,8 @@ "content": "$\\mathrm{C}_n^0+2\\mathrm{C}_n^1+2^2\\mathrm{C}_n^2+\\cdots +2^n\\mathrm{C}_n^n$的值为\\bracket{20}.\n\\fourch{$2^n$}{$2^{n-1}$}{$3^n$}{$3^{n-1}$}", "objs": [], "tags": [ - "第八单元" + "第八单元", + "二项式定理" ], "genre": "选择题", "ans": "", @@ -191290,7 +191886,8 @@ "content": "$101^{10}-1$的末尾连续零的个数是\\bracket{20}.\n\\fourch{$1$}{$2$}{$3$}{$4$}", "objs": [], "tags": [ - "第八单元" + "第八单元", + "二项式定理" ], "genre": "选择题", "ans": "", @@ -191311,7 +191908,8 @@ "content": "若$\\mathrm{C}_n^0(x+1)^n-\\mathrm{C}_n^1(x+1)^{n-1}+\\mathrm{C}_n^2(x+1)^{n-2}-\\cdots +(-1)^n\\mathrm{C}_n^n=a_0x^n+a_1x^{n-1}+\\cdots +a_{n-1}x+a_n$, 则$a_1+a_2+\\cdots +a_n=$\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "二项式定理" ], "genre": "填空题", "ans": "", @@ -191420,7 +192018,8 @@ "content": "求满足$\\{a,b\\}\\subset A\\subseteq \\{a,b,c,d,e,f,g\\}$的集合$A$的个数.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "加法原理与乘法原理" ], "genre": "解答题", "ans": "", @@ -191441,7 +192040,8 @@ "content": "设集合$A=\\{0,2,5,7,9\\}$, 从集合$A$中任取两个元素相乘, 它们的积组成集合$B$, 求集合$B$的子集的个数.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "加法原理与乘法原理" ], "genre": "解答题", "ans": "", @@ -191462,7 +192062,8 @@ "content": "求和: $\\mathrm{C}_{100}^0+4\\mathrm{C}_{100}^1+7\\mathrm{C}_{100}^2+\\cdots +(3n-2)\\mathrm{C}_{100}^{n-1}+\\cdots +298\\mathrm{C}_{100}^{99}+301\\mathrm{C}_{100}^{100}$($n\\in \\mathbf{N}$, $1\\le n\\le 101$).", "objs": [], "tags": [ - "第八单元" + "第八单元", + "二项式定理" ], "genre": "解答题", "ans": "", @@ -191483,7 +192084,8 @@ "content": "设$a_0, a_1, a_2, \\cdots,a_n$是等差数列, 求证: $a_0+\\mathrm{C}_n^1a_1+\\mathrm{C}_n^2a_2+\\cdots +\\mathrm{C}_n^na_n=(a_0+a_n)\\cdot 2^{n-1}$.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "二项式定理" ], "genre": "解答题", "ans": "", @@ -191504,7 +192106,8 @@ "content": "若$n$为奇数, 求$7^n+\\mathrm{C}_n^1\\cdot 7^{n-1}+\\mathrm{C}_n^2\\cdot 7^{n-2}+\\mathrm{C}_n^37^{n-3}+\\cdots +\\mathrm{C}_n^{n-2}\\cdot 7^2+\\mathrm{C}_n^{n-1}\\cdot 7$被$9$除所得的余数.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "二项式定理" ], "genre": "解答题", "ans": "", @@ -191525,7 +192128,8 @@ "content": "求$47^{13}$被$5$除的余数.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "二项式定理" ], "genre": "解答题", "ans": "", @@ -191546,7 +192150,8 @@ "content": "求$91^{92}$除以$8$所得的余数.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "二项式定理" ], "genre": "解答题", "ans": "", @@ -191567,7 +192172,8 @@ "content": "求证: $3^{2n}-8n-1$($n\\in \\mathbf{N}$)能被$64$整除.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "二项式定理" ], "genre": "解答题", "ans": "", @@ -191590,7 +192196,8 @@ "content": "求证: 数列$65,65\\times 66, 65\\times 66^2, 65\\times 66^3, \\cdots, 65\\times 66^{48}, 65\\times 66^{49}$之和必能被$67$整除.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "二项式定理" ], "genre": "解答题", "ans": "", @@ -191611,7 +192218,8 @@ "content": "已知$2^{n+2}\\times 3^n+5n-a$($n\\in \\mathbf{N}$)能被$25$整除, 求$a$的最小正值.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "二项式定理" ], "genre": "解答题", "ans": "", @@ -191632,7 +192240,8 @@ "content": "求$x^{10}-3$除以$(x-1)^2$所得的余式.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "二项式定理" ], "genre": "解答题", "ans": "", @@ -191653,7 +192262,8 @@ "content": "求证: 当$n\\ge 3$, $n\\in \\mathbf{N}$时, $n^{n-1}-1$能被$(n-1)^2$整除.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "二项式定理" ], "genre": "解答题", "ans": "", @@ -191674,7 +192284,8 @@ "content": "设$(x-2)^8=a_8x^8+a_7x^7+\\cdots +a_1x+a_0$, 求$a_8+a_6+a_4+a_2$.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "二项式定理" ], "genre": "解答题", "ans": "", @@ -191717,7 +192328,8 @@ "content": "已知$(3-x)^n=a_0+a_1x+a_2x^2+a_3x^3+\\cdots +a_nx^n$, 求$a_1+2a_2+2^2a_3+\\cdots +2^{n-1}a_n$.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "二项式定理" ], "genre": "解答题", "ans": "", @@ -191738,7 +192350,8 @@ "content": "求证: $\\mathrm{C}_n^0\\mathrm{C}_n^1+\\mathrm{C}_n^1\\mathrm{C}_n^2+\\cdots +\\mathrm{C}_n^{n-1}\\mathrm{C}_n^n=\\dfrac{(2n)!}{(n-1)!(n+1)!}$.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "二项式定理" ], "genre": "解答题", "ans": "", @@ -191761,7 +192374,8 @@ "content": "求证: $\\mathrm{C}_n^0\\mathrm{C}_m^p+\\mathrm{C}_n^1\\mathrm{C}_m^{p-1}+\\cdots +\\mathrm{C}_n^p\\mathrm{C}_m^0=\\mathrm{C}_{m-n}^p$($p\\le m,n$).", "objs": [], "tags": [ - "第八单元" + "第八单元", + "二项式定理" ], "genre": "解答题", "ans": "", @@ -191782,7 +192396,8 @@ "content": "利用$k\\mathrm{C}_n^k=n\\mathrm{C}_{n-1}^{k-1}$, 求证: $\\mathrm{C}_n^1+2\\mathrm{C}_n^2+3\\mathrm{C}_n^3+\\cdots +n\\mathrm{C}_n^n=n\\cdot 2^{n-1}$.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "二项式定理" ], "genre": "解答题", "ans": "", @@ -191806,7 +192421,8 @@ "content": "利用$k\\mathrm{C}_n^k=n\\mathrm{C}_{n-1}^{k-1}$, 求证: $\\mathrm{C}_n^1-2\\mathrm{C}_n^2+3\\mathrm{C}_n^3+\\cdots +(-1)^{n-1}n\\mathrm{C}_n^n=0$($n\\ge 2$, $n\\in \\mathbf{N}$).", "objs": [], "tags": [ - "第八单元" + "第八单元", + "二项式定理" ], "genre": "解答题", "ans": "", @@ -191829,7 +192445,8 @@ "content": "利用$k\\mathrm{C}_n^k=n\\mathrm{C}_{n-1}^{k-1}$, 求证: $\\mathrm{C}_n^0+2\\mathrm{C}_n^1+3\\mathrm{C}_n^2+\\cdots +(n+1)\\mathrm{C}_n^n=(n+2)\\cdot 2^{n-1}$.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "二项式定理" ], "genre": "解答题", "ans": "", @@ -191852,7 +192469,8 @@ "content": "已知$n\\in \\mathbf{N}$, $n\\ge 2$, 求证: $2^n>1+2+\\cdots +n$.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "二项式定理" ], "genre": "解答题", "ans": "", @@ -191873,7 +192491,8 @@ "content": "求证: $3^n>2^{n-1}(n+2)$($n>2$, $n\\in \\mathbf{N}$).", "objs": [], "tags": [ - "第八单元" + "第八单元", + "二项式定理" ], "genre": "解答题", "ans": "", @@ -191894,7 +192513,8 @@ "content": "已知正数$a,b,c$满足$a+b+c=abc$, 求证: $a^n+b^n+c^n>3(1+\\dfrac n2)$($n\\in \\mathbf{N}$).", "objs": [], "tags": [ - "第八单元" + "第八单元", + "二项式定理" ], "genre": "解答题", "ans": "", @@ -191915,7 +192535,8 @@ "content": "利用数学归纳法证明: $(\\dfrac n2)^n>n!$($n\\in \\mathbf{N}$且$n\\ge 6$).", "objs": [], "tags": [ - "第八单元" + "第八单元", + "二项式定理" ], "genre": "解答题", "ans": "", @@ -191958,7 +192579,8 @@ "content": "若实数$x,y$满足$x+y=1$, 求证: $x^5+y^5\\ge \\dfrac 1{16}$.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "二项式定理" ], "genre": "解答题", "ans": "", @@ -191979,7 +192601,8 @@ "content": "已知: $|x|<1$, $n\\in \\mathbf{N}$, $n\\ge 2$, 求证: $(1-x)^n+(1+x)^n<2^n$.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "二项式定理" ], "genre": "解答题", "ans": "", @@ -192000,7 +192623,8 @@ "content": "计算: $\\mathrm{C}_{21}^0-\\mathrm{C}_{21}^2+\\mathrm{C}_{21}^4-\\mathrm{C}_{21}^6+\\mathrm{C}_{21}^8-\\mathrm{C}_{21}^{10}+\\mathrm{C}_{21}^{12}-\\mathrm{C}_{21}^{14}+\\mathrm{C}_{21}^{16}-\\mathrm{C}_{21}^{18}+\\mathrm{C}_{21}^{20}$.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "二项式定理" ], "genre": "解答题", "ans": "", @@ -192021,7 +192645,8 @@ "content": "求证: $1+\\mathrm{C}_n^1\\cos \\alpha +\\mathrm{C}_n^2\\cos 2\\alpha +\\cdots +\\mathrm{C}_n^n\\cos n\\alpha =2^n\\cos ^n(\\dfrac{\\alpha }2)\\cdot \\cos \\dfrac{n\\alpha }2$,\n$\\mathrm{C}_n^1\\sin \\alpha +\\mathrm{C}_n^2\\sin 2\\alpha +\\cdots +\\mathrm{C}_n^n\\sin n\\alpha =2^n\\cos ^n(\\dfrac{\\alpha }2)\\sin \\dfrac{n\\alpha }2$.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "二项式定理" ], "genre": "解答题", "ans": "", @@ -192042,7 +192667,8 @@ "content": "设$a_n=1+q+q^2+\\cdots +q^{n-1}$($n\\in \\mathbf{N}$, $q\\ne \\pm 1$), $A_n=a_1\\mathrm{C}_n^1+a_2\\mathrm{C}_n^2+\\cdots +a_n\\mathrm{C}_n^n$.\\\\\n(1) 用$q,n$表示$A_n$;\\\\\n(2) 当$-3\\dfrac 1{10}$, 试比较$A_n$和$B_n$的大小, 并证明你的结论.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "二项式定理" ], "genre": "解答题", "ans": "", @@ -192084,7 +192711,8 @@ "content": "$6$人按下列要求分组, 各有多少种分法.\\\\\n(1) 分成人数为$2$, $4$的两组;\\\\\n(2) 分成人数相等的两组;\\\\\n(3) 平均分成两组分别去植树和扫地.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "解答题", "ans": "", @@ -192105,7 +192733,8 @@ "content": "某校以单循环制方法进行排球比赛, 其中有两个班级各比赛了$3$次后, 不再参加比赛, 这样一共进行了$84$场比赛, 问: 开始有多少班级参加比赛?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "解答题", "ans": "", @@ -192126,7 +192755,8 @@ "content": "红、黄、绿$3$种颜色的卡片分别写有$A$, $B$, $C$, $D$, $E$E字母各一张, 每次取出$5$张, 要求字母各不相同、$3$种颜色齐全的取法有多少种?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "解答题", "ans": "", @@ -192147,7 +192777,8 @@ "content": "设$n$为偶数, 从$1, 2, \\cdots, n$中选$3$数使之不构成等差数列, 问: 这样的选法有多少种?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "加法原理与乘法原理" ], "genre": "解答题", "ans": "", @@ -192168,7 +192799,8 @@ "content": "设集合$P=\\{a_1,a_2,\\cdots ,a_n\\}$, 在$P$中取子集$A_1$, $A_2$, $A_3$, 使$A_1\\cap A_2\\cap A_3=\\varnothing$, 这样子集的集合$\\{A_1,A_2,A_3\\}$共有多少个?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "加法原理与乘法原理" ], "genre": "解答题", "ans": "", @@ -192189,7 +192821,8 @@ "content": "如图, 有纵路$10$条, 横路$2$条, 从$A$沿道路行走到$B$, 规定行走中不得重走已走过的路, 共有多少种不同的走法?\n\\begin{center}\n \\begin{tikzpicture}\n \\draw (0,0) node [below] {$A$} -- (9,0) node [below] {$B$};\n \\draw (0,1) -- (9,1);\n \\foreach \\i in {0,1,...,9} {\\draw (\\i,0) -- (\\i,1);};\n \\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "解答题", "ans": "", @@ -192210,7 +192843,8 @@ "content": "由$1$分, $2$分, $5$分, $1$角, $2$角, $5$角, $1$元, $2$元, $5$元, $10$元人民币各一张, 可组成多少种不同的币值?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "加法原理与乘法原理" ], "genre": "解答题", "ans": "", @@ -192231,7 +192865,8 @@ "content": "壹分币$3$枚、贰角币$6$张、拾元币$4$张, 可以组成多少种不同的币值?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "加法原理与乘法原理" ], "genre": "解答题", "ans": "", @@ -192252,7 +192887,8 @@ "content": "求$21600$的正约数的个数($1$和$21600$也是约数)及所有约数之和.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "加法原理与乘法原理" ], "genre": "解答题", "ans": "", @@ -192273,7 +192909,9 @@ "content": "设自然数$N=\\{1,2,3,\\cdots\\}$的子集中含有$4$个元素的子集的个数记为$m$, 且这$m$个集合中所有元素之和为$\\dfrac 1{12}\\mathrm{P}_{100}^5$, 求$m$.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列", + "加法原理与乘法原理" ], "genre": "解答题", "ans": "", @@ -192294,7 +192932,8 @@ "content": "有$11$名工人, 其中$5$名只会做钳工, $4$名只会做车工, $2$名既会做钳工, 又会做车工, 今要选$4$名车工、$4$名钳工, 有多少种不同的选法?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "解答题", "ans": "", @@ -192315,7 +192954,8 @@ "content": "设$(1+x+x^2)^n=a_0+a_1x+a_2x^2+\\cdots +a_{2n}x^{2n}$, 求$a_0+a_2+a_4+\\cdots +a_{2n}$的值.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "二项式定理" ], "genre": "解答题", "ans": "", @@ -192380,7 +193020,8 @@ "content": "求证: $(3+\\sqrt 7)^n$($n\\in \\mathbf{N}$, $n\\ge 2$)的整数部分为奇数.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "二项式定理" ], "genre": "解答题", "ans": "", @@ -229125,7 +229766,8 @@ "content": "一种旅行包上的号码锁有三个拨号盘, 每个拨号盘上有从$0$到$9$的$10$个数字, 这三个拨号盘可组成多少种不同的三位数号码?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "加法原理与乘法原理" ], "genre": "解答题", "ans": "", @@ -229146,7 +229788,8 @@ "content": "一个商场共有$9$个出入口, 若某人在进出商场时不要走同一个出入口, 则他一次进出商场共有多少种不同的进出法?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "加法原理与乘法原理" ], "genre": "解答题", "ans": "", @@ -229167,7 +229810,8 @@ "content": "已知$a\\in \\{1,3,5,7\\}$, $b\\in \\{2,4,6,8\\}$, 在平面直角坐标系中, 直线方程$ax+by+1=0$可以表示多少条直线?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "加法原理与乘法原理" ], "genre": "解答题", "ans": "", @@ -229188,7 +229832,8 @@ "content": "为了提高产品质量控制生产过程的温度、材料处理的时间和添加剂的剂量, 为此工厂进行生产试验. 试验控制温度有$150^\\circ\\text{C}$、$160^\\circ\\text{C}$和$170^\\circ\\text{C}$三种, 材料处理的时间有$10$分钟、$12$分钟两种, 添加剂的剂量有$2$克, $4$克和$6$克三种, 共需要做多少次试验?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "加法原理与乘法原理" ], "genre": "解答题", "ans": "", @@ -229209,7 +229854,8 @@ "content": "$(a_1+a_2+a_3)(b_1+b_2+b_3+b_4)(c_1+c_2)$展开后共有多少项?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "加法原理与乘法原理" ], "genre": "解答题", "ans": "", @@ -229232,7 +229878,8 @@ "content": "某个学校食堂准备了$5$种素菜、$3$种荤菜和$3$种汤, 取一种素菜、一种荤菜、一种汤配成一套菜, 这个学校食堂可以有多少套不同的菜?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "加法原理与乘法原理" ], "genre": "解答题", "ans": "", @@ -229253,7 +229900,8 @@ "content": "用$1$、$2$、$3$、$4$、$5$这五个数字可以组成多少个无重复数字的三位数的奇数?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "加法原理与乘法原理" ], "genre": "解答题", "ans": "", @@ -229274,7 +229922,8 @@ "content": "要把$4$封信投入$3$个信箱, 共有多少种不同的投法? (允许将信全部或部分投入某一个信箱)", "objs": [], "tags": [ - "第八单元" + "第八单元", + "加法原理与乘法原理" ], "genre": "解答题", "ans": "", @@ -229297,7 +229946,8 @@ "content": "用$0$、$1$、$2$、$3$、$4$、$5$这六个数字可以组成多少个数字不重复的三位数?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "加法原理与乘法原理" ], "genre": "解答题", "ans": "", @@ -229321,7 +229971,8 @@ "content": "用$0$、$1$、$2$、$3$、$4$、$5$这六个数字可以组成多少个三位数?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "加法原理与乘法原理" ], "genre": "解答题", "ans": "", @@ -229345,7 +229996,8 @@ "content": "已知集合$M=\\{-3,-2,-1,0,1,2\\}$, 点$P(a,b) $在直角坐标平面上, 且$a, b\\in M$.\\\\\n(1) 平面上共有多少个满足条件的点$P$?\\\\\n(2) 有多少个点$P$在第二象限内?\\\\\n(3) 有多少个点$P$不在直线$y=x$上?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "加法原理与乘法原理" ], "genre": "解答题", "ans": "", @@ -229366,7 +230018,8 @@ "content": "从$15$件不同的礼品中取出$4$件分送给$4$个学生, 共有多少种不同的送法?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "解答题", "ans": "", @@ -229387,7 +230040,8 @@ "content": "从$5$名运动员中选出$3$名参加乒乓球团体比赛, 并排定他们的出场顺序, 有多少种不同的方法?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "解答题", "ans": "", @@ -229408,7 +230062,8 @@ "content": "从若干种不同的盆景中选出$2$种摆放在阳台的左右两侧, 如果想要有$30$种不同的选法, 那么最少要准备多少种不同的盆景?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "解答题", "ans": "", @@ -229429,7 +230084,8 @@ "content": "从$2,3,4,5,7,11$这六个数字中选出$2$个数字作为分子和分母, 共能组成多少个大小不同的分数?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "解答题", "ans": "", @@ -229450,7 +230106,8 @@ "content": "从$6$名志愿者中选出$4$人分别从事翻译、导游、导购、保洁工作, 其中甲、乙两人不能从事翻译工作, 选派志愿者的方案共有多少种?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "解答题", "ans": "", @@ -229471,7 +230128,8 @@ "content": "求下列各式中$n$($n\\in \\mathbf{N}^*$)的值.\\\\\n(1) $\\mathrm{P}_{2n}^3=11\\mathrm{P}_n^3$;\\\\\n(2) $\\mathrm{P}_n^5+\\mathrm{P}_n^4=4\\mathrm{P}_n^3$;\\\\\n(3) $\\mathrm{P}_n^3=n\\mathrm{P}_3^3$.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "解答题", "ans": "", @@ -229492,7 +230150,8 @@ "content": "用$1$、$2$、$3$、$4$、$5$、$6$能组成多少个没有重复数字且大于$500$的三位数?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "解答题", "ans": "", @@ -229515,7 +230174,8 @@ "content": "用$1$、$2$、$3$、$4$、$5$、$6$能组成多少个没有重复数字且小于$500$的三位数?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "解答题", "ans": "", @@ -229538,7 +230198,8 @@ "content": "已知$\\mathrm{P}_{10}^m=10\\times 9\\times \\cdots \\times 5$, 求正整数$m$的值.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "解答题", "ans": "", @@ -229559,7 +230220,8 @@ "content": "从$6$名学生中任选$3$人分别担任语文、数学、英语课代表, 其中学生甲不能担任数学课代表, 共有多少种不同的选法?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "解答题", "ans": "", @@ -229582,7 +230244,8 @@ "content": "$5$名学生站成一排, 其中甲学生不能站在排头的不同站法有多少种?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "解答题", "ans": "", @@ -229603,7 +230266,8 @@ "content": "$4$名教师、$3$名男生、$2$名女生排成一排, 要求$3$名男生排在一起, $2$名女生排在一起, 共有多少种不同的排队方法?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "解答题", "ans": "", @@ -229624,7 +230288,8 @@ "content": "用$0$、$1$、$2$、$3$、$4$、$5$这六个数字可以组成多少个没有重复数字的四位奇数?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "解答题", "ans": "", @@ -229648,7 +230313,8 @@ "content": "用$0$到$9$这十个数可以组成多少个没有重复数字的四位数?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "解答题", "ans": "", @@ -229671,7 +230337,8 @@ "content": "已知甲、乙、丙等$7$人站成一排, 求分别按下列要求排队各有多少种不同的排法;\\\\\n(1) 甲、乙都与丙相邻;\\\\\n(2) 甲、乙之间有且只有$1$人.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "解答题", "ans": "", @@ -229692,7 +230359,8 @@ "content": "化简: $\\dfrac 1{2!}+\\dfrac 2{3!}+\\dfrac 3{4!}+\\cdots +\\dfrac{n-1}n$($n\\in \\mathbf{N}^*$, $n\\ge 2$).", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "解答题", "ans": "", @@ -229715,7 +230383,8 @@ "content": "已知抛物线方程为$y=ax^2+bx+c$, 集合$M=\\{-2,-1,0,1,2,3,4\\}$, $a,b,c\\in M$, 且$a,b,c$两两不相等, 满足条件的抛物线中, 过原点的抛物线有多少条?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "解答题", "ans": "", @@ -229736,7 +230405,8 @@ "content": "求证: $\\mathrm{P}_1^1+2\\mathrm{P}_2^2+3\\mathrm{P}_3^3+\\cdots +n\\mathrm{P}_n^n=\\mathrm{P}_{n+1}^{n+1}-1$($n\\in \\mathbf{N}^*$).", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "解答题", "ans": "", @@ -229757,7 +230427,8 @@ "content": "乒乓球队的$10$名队员中有$3$名主力队员, 派$5$名队员参加比赛, 其中, $3$名主力队员要安排在第一、三、五位置, 其余$7$名队员中的$2$名要安排在第二、四位置, 共有多少种不同的安排方法?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "解答题", "ans": "", @@ -229778,7 +230449,8 @@ "content": "用$0$、$6$、$8$这三个数字可组成多少个没有重复数字的整数?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "解答题", "ans": "", @@ -229799,7 +230471,8 @@ "content": "用$0$到$9$这十个数字可组成多少个能被$5$整除的无重复数字的二位数?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "解答题", "ans": "", @@ -229820,7 +230493,8 @@ "content": "某班的新年联欢会原定的$5$个节目已排成节目单, 开始演出前又增加了$2$个新节目, 如果将这两个新节目插入原节目单中, 那么有多少种不同的插法?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "解答题", "ans": "", @@ -229841,7 +230515,8 @@ "content": "用$0$到$9$这十个数字, 可组成多少个没有重复数字的四位数的偶数?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "解答题", "ans": "", @@ -229864,7 +230539,8 @@ "content": "有$A,B,C,D,E$五列火车停在某车站并行的$5$条火车轨道上, 如果快车$A$不能停在第$3$道上, 慢车$B$不能停在第$1$道上, 那么这五列火车的停车方法有多少种?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "解答题", "ans": "", @@ -229885,7 +230561,8 @@ "content": "某班级周一的课表要排入政治、语文、数学、物理、化学、体育共$6$门学科, 如果第一节课不排体育课, 最后一节课不排数学课, 那么共有多少种不同的排法?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "解答题", "ans": "", @@ -229906,7 +230583,8 @@ "content": "用$0$到$5$这六个数字可组成无重复数字的四位数的偶数, 且这个偶数的百位、十位上都是奇数, 满足条件的数共有多少个?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "解答题", "ans": "", @@ -229927,7 +230605,8 @@ "content": "将$8$个相同的小球放入编号为$1$、$2$、$3$的三个盒内, 要求每个盒子的球数不小于它的编号数, 共有多少种不同的放法?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "加法原理与乘法原理" ], "genre": "解答题", "ans": "", @@ -229948,7 +230627,8 @@ "content": "用$1$、$2$、$3$、$4$、$5$可组成多少个无重复数字且比$13245$大的五位数?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "解答题", "ans": "", @@ -229969,7 +230649,9 @@ "content": "试确定下列问题是排列问题还是组合问题.\\\\\n(1) $3$本不同的书借给甲、乙、丙$3$名学生, 每人$1$本, 有多少种不同的借法?\\\\\n(2) 从$10$本书中任意取$5$本赠送给$1$名学生, 有多少种不同的送法?\\\\\n(3) 从$15$人中选$3$人去参加数学竞赛, 有多少种不同的选法?\\\\\n(4) 从$15$人中选$3$人分别参加数学、物理、化学竞赛, 有多少种不同的选法?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合", + "排列" ], "genre": "解答题", "ans": "", @@ -229990,7 +230672,8 @@ "content": "某项测试共有两组试题. 要求从第一组$10$个问题中选择$8$个, 从第二组$5$个问题中选择$4$个, 要完成这项测试有多少种不同的选择试题的方法?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "解答题", "ans": "", @@ -230011,7 +230694,8 @@ "content": "已知$100$件产品中有$2$件次品如果从这些产品中任取$5$件, 那么其中恰好有$2$件次品的取法有多少种?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "解答题", "ans": "", @@ -230032,7 +230716,8 @@ "content": "某班级共有$25$名团员, 其中$10$名男团员, $15$名女团员.\\\\\n(1) 如果从中推选$2$名男团员和$3$名女团员参加团代会, 那么有多少种不同的推选方法?\\\\\n(2) 如果从中推选$2$名男团员和$3$名女团员组成团支部分别担任不同职务, 那么有多少种不同的推选方法?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "解答题", "ans": "", @@ -230053,7 +230738,8 @@ "content": "以某个圆周上的$10$个点为顶点, 可以作多少个三角形?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "解答题", "ans": "", @@ -230074,7 +230760,8 @@ "content": "求下列各式中$n$($n\\in \\mathbf{N}^*$)的值:\\\\\n(1) $\\mathrm{C}_n^5+\\mathrm{C}_n^6=\\mathrm{C}_{n+1}^3$;\\\\\n(2) $\\mathrm{C}_{n+1}^{n-1}=\\dfrac 7{15}\\mathrm{P}_{n+1}^3$.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "解答题", "ans": "", @@ -230095,7 +230782,8 @@ "content": "求证: $\\mathrm{C}_n^m=\\dfrac{m+1}{n+1}\\mathrm{C}_{n+1}^{m+1}$($n,m\\in \\mathbf{N}^*$, $n\\ge m$).", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "解答题", "ans": "", @@ -230116,7 +230804,8 @@ "content": "计算: $\\mathrm{C}_3^0+\\mathrm{C}_4^1+\\mathrm{C}_5^2+\\cdots +\\mathrm{C}_{20}^{17}$.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "解答题", "ans": "", @@ -230143,7 +230832,8 @@ "content": "从$8$名男运动员与$7$名女运动员中选出$5$名男运动员与$5$名女运动员组成一个运动队, 不同的选法共有多少种?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "解答题", "ans": "", @@ -230164,7 +230854,8 @@ "content": "要从$6$名男学生与$6$名女学生中选出$2$名男学生与$2$名女学生组成一个学习小组, 共有多少种不同的选法?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "解答题", "ans": "", @@ -230185,7 +230876,8 @@ "content": "已知平面上共有$10$个点, 其中有$4$个点在一条直线上, 除此之外再没有三点共线, 以这$10$个点为顶点能组成多少不同的三角形?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "解答题", "ans": "", @@ -230206,7 +230898,8 @@ "content": "(1) 计算$\\mathrm{C}_2^0+\\mathrm{C}_2^1+\\mathrm{C}_2^2$;\\\\\n(2) 计算: $\\mathrm{C}_3^0+\\mathrm{C}_3^1+\\mathrm{C}_3^2+\\mathrm{C}_3^3$;\\\\\n(3) 猜想$\\mathrm{C}_n^0+\\mathrm{C}_n^1+\\mathrm{C}_n^2+\\cdots +\\mathrm{C}_n^{n-1}+\\mathrm{C}_n^n(n\\in \\mathbf{N}^*) $的值, 并证明你的结果;\\\\\n(4) 你能否利用第(3)题来求一个集合的子集的个数? 为什么?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "解答题", "ans": "", @@ -230227,7 +230920,8 @@ "content": "用一组$5$条平行线与另一组$4$条平行线共可围成多少个平行四边形?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "解答题", "ans": "", @@ -230248,7 +230942,8 @@ "content": "已知$\\dfrac{\\mathrm{C}_{2n}^{n-1}}{\\mathrm{C}_2^n(n-1)}=\\dfrac{56}{15}$, 求正整数$n$的值.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "解答题", "ans": "", @@ -230269,7 +230964,8 @@ "content": "从$3$本不同的语文书、$4$本不同的数学书和$3$本不同的物理书中取出$4$本书, 且要求三种书都有共有多少种不同的取法?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "解答题", "ans": "", @@ -230290,7 +230986,8 @@ "content": "从$5$名女学生和$4$名男学生中选出$4$人担任$4$种不同的工作, 且要求选出的$4$人中男女学生都有, 共有多少种不同的选法?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "解答题", "ans": "", @@ -230311,7 +231008,8 @@ "content": "已知集合$AB$都含有$12$个元素, $A\\cap B$含有$4$个元素, 集合$C$含有$3$个元素, 且$C\\subset A\\cup B,C\\cap B\\ne \\varnothing$, 求满足条件的集合$C$的个数.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "解答题", "ans": "", @@ -230332,7 +231030,8 @@ "content": "某旅游团要从$8$个风景点中选$2$个风景点作为当天的旅游地, 求分别满足以下条件的选法的种数.\\\\\n(1) 甲乙风景中至少选一个;\\\\\n(2) 甲乙风景点中至多选一个;\\\\\n(3) 甲乙风景点中必须选一个, 而且只能选一个.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "解答题", "ans": "", @@ -230375,7 +231074,8 @@ "content": "化简:\\\\\n(1) $(1+\\sqrt x)^5+(1-\\sqrt x)^5$;\\\\\n(2) $(2x+y)^4-(2x-y)^4$.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "二项式定理" ], "genre": "解答题", "ans": "", @@ -230506,7 +231206,8 @@ "content": "求证: $2^n-\\mathrm{C}_n^1\\cdot 2^{n-1}+\\mathrm{C}_n^2\\cdot 2^{n-2}+\\cdots +\\mathrm{C}_n^{n-1}\\cdot 2+(-1)^n=1$.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "二项式定理" ], "genre": "解答题", "ans": "", @@ -230527,7 +231228,8 @@ "content": "$\\mathrm{C}_n^1+3\\mathrm{C}_n^2+9\\mathrm{C}_n^3+\\cdots +3^{n-1}\\mathrm{C}_n^n$等于\\bracket{20}.\n\\fourch{$4^n$}{$\\dfrac{4^n}3$}{$\\dfrac{4^n}3-1$}{$\\dfrac{4^n-1}3$}", "objs": [], "tags": [ - "第八单元" + "第八单元", + "二项式定理" ], "genre": "选择题", "ans": "", @@ -230548,7 +231250,8 @@ "content": "已知$n$为大于$1$的自然数, 证明: $(1+\\dfrac 1n)^n>2$.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "二项式定理" ], "genre": "解答题", "ans": "", @@ -230591,7 +231294,8 @@ "content": "选择题:\n$\\mathrm{C}_{100}^0-\\mathrm{C}_{100}^2+\\mathrm{C}_{100}^4-\\cdots +\\mathrm{C}_{100}^{98}+\\mathrm{C}_{100}^{100}$等于\\bracket{20}.\n\\fourch{$-2^{50}$}{$0$}{$1$}{$2^{50}$}", "objs": [], "tags": [ - "第八单元" + "第八单元", + "二项式定理" ], "genre": "选择题", "ans": "", @@ -230678,7 +231382,8 @@ "content": "求$77^{77}-15$除以$19$的余数.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "二项式定理" ], "genre": "解答题", "ans": "", @@ -230701,7 +231406,8 @@ "content": "求证: $2^{6n-3}+3^{2n-1}$能被$11$整除.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "二项式定理" ], "genre": "解答题", "ans": "", @@ -230722,7 +231428,8 @@ "content": "已知$(x+1)^n=x^n+\\cdots +ax^3+bx^2+cx+1$($n\\in \\mathbf{N}^*$), 且$a:b=3:1$, 求$c$的值.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "二项式定理" ], "genre": "解答题", "ans": "", @@ -230743,7 +231450,8 @@ "content": "某学生要从$2$本科技书、$2$本政治书和$3$本文艺书中任取一本书, 共有\\blank{50}种不同的取法.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "加法原理与乘法原理" ], "genre": "填空题", "ans": "", @@ -230764,7 +231472,8 @@ "content": "如果将$3$名男学生与$2$名女学生排成一排, 且$2$名女生不排在相邻位置上, 那么不同排法的种数是\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "填空题", "ans": "", @@ -230785,7 +231494,8 @@ "content": "计划在某画廊展出$10$幅不同的画, 其中$1$幅为水彩画, $4$幅为油画, $5$幅为国画, 排成一行陈列, 如果同一品种的画必须排在一起, 并且水彩不能排在两端, 那么陈列方式有\t\\blank{50}种.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "填空题", "ans": "", @@ -230806,7 +231516,8 @@ "content": "用数字$0$、$1$、$2$、$3$、$4$、$5$可组成没有重复数字的六位数, 其中数字$2$、$4$排在相邻数位上, 满足条件的六位数共有\\blank{50}个.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "填空题", "ans": "", @@ -230849,7 +231560,8 @@ "content": "若把$4$只不同颜色的球放人$3$个不同的袋内, 则不同的放法的种数是\\bracket{20}.\n\\fourch{$4^3$}{$3^4$}{$\\mathrm{P}_4^3$}{$\\mathrm{C}_4^3$}", "objs": [], "tags": [ - "第八单元" + "第八单元", + "加法原理与乘法原理" ], "genre": "选择题", "ans": "", @@ -230870,7 +231582,9 @@ "content": "若$\\mathrm{C}_n^3=12\\mathrm{P}_n^1$, 则$n$的值为\\bracket{20}.\n\\fourch{$3$}{$5$}{$7$}{$10$}", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合", + "排列" ], "genre": "选择题", "ans": "", @@ -230891,7 +231605,8 @@ "content": "语文兴趣小组有学生$10$人, 从中选派$3$人参加诗歌朗诵会, 不同的选派方法的种数是\\bracket{20}.\n\\fourch{$720$}{$360$}{$240$}{$120$}", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "选择题", "ans": "", @@ -230912,7 +231627,8 @@ "content": "用$1$、$2$、$3$、$4$、$5$这五个数字可以组成比$20000$大, 且百位数不是$3$的没有重复数字的五位数, 满足条件的五位数共有\\bracket{20}.\n\\fourch{$96$个}{$78$个}{$77$个}{$64$个}", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "选择题", "ans": "", @@ -230933,7 +231649,8 @@ "content": "某市工商局会同商检局对$35$种商品进行抽样检查, 鉴定结果为其中有$5$种是不合格商品, 现从这$35$种商品中任取$3$种, 至少有$2$种不合格商品的取法种数是\\bracket{20}.\n\\fourch{$\\mathrm{C}_5^3+\\mathrm{C}_5^2\\mathrm{C}_{30}^1$}{$\\mathrm{P}_5^3+\\mathrm{P}_5^2\\mathrm{P}_{30}^1$}{$\\mathrm{C}_5^2\\mathrm{C}_{30}^1$}{$\\mathrm{P}_5^2\\mathrm{P}_{30}^1$}", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "选择题", "ans": "", @@ -230976,7 +231693,8 @@ "content": "某学生邀请$10$位同学中的$6$位参加一个生日聚会, 其中$2$位同学要么都邀请, 要么都不邀请, 共有多少种邀请方法?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "解答题", "ans": "", @@ -230997,7 +231715,8 @@ "content": "某次篮球赛预赛分成$3$个赛区进行, 第一赛区男队、女队各$9$队, 第二赛区男队、女队各$10$队, 第三赛区男队$9$队, 女队$10$队, 各赛区男队、女队各取前$4$名参加决赛. 预赛、决赛都采用单循环制比赛, 一共需要进行多少场比赛?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "解答题", "ans": "", @@ -231040,7 +231759,8 @@ "content": "关于$x$的方程$\\mathrm{C}_{34}^{x^2-2x}=\\mathrm{C}_{34}^{5x-6}$的解集是\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "填空题", "ans": "", @@ -231061,7 +231781,8 @@ "content": "$6$个人排成一列, 其中甲乙两人之间至少有两个人的不同排法种数是\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "填空题", "ans": "", @@ -231082,7 +231803,8 @@ "content": "由数字$1$、$2$、$3$、$4$组成没有重复数字的不同自然数的个数是\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "填空题", "ans": "", @@ -231103,7 +231825,8 @@ "content": "若$m\\in \\{2,5,7,8\\}$, $n\\in\\{1,3,4,6\\}$, 则方程$\\dfrac{x^2}m+\\dfrac{y^2}n=1$表示焦点在$x$轴上的椭圆有\\blank{50}个.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "加法原理与乘法原理" ], "genre": "填空题", "ans": "", @@ -231146,7 +231869,8 @@ "content": "已知在$100$件产品中有$3$件是次品, 如果从中任意抽取$5$件, 那么其中至多有$2$件次品的抽法的种数是\\bracket{20}.\n\\fourch{$\\mathrm{C}_3^2\\mathrm{C}_{97}^3$}{$\\mathrm{C}_{100}^5\\mathrm{C}_3^2$}{$\\mathrm{C}_{100}^5\\mathrm{C}_3^2\\mathrm{C}_{97}^2$}{$\\mathrm{C}_3^2\\mathrm{C}_{97}^2+\\mathrm{C}_3^1\\mathrm{C}_{97}^4$}", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "选择题", "ans": "", @@ -231167,7 +231891,8 @@ "content": "从$10$名男学生和$12$名女学生中各选$3$名排成一列, 其中男、女相间排成一列的不同排法的种数是\\bracket{20}.\n\\fourch{$2\\mathrm{P}_{10}^3\\mathrm{P}_{12}^3$}{$\\mathrm{P}_{10}^3\\mathrm{P}_{12}^3$}{$\\mathrm{C}_4^3\\mathrm{P}_{10}^3\\mathrm{P}_{12}^3$ }{$\\mathrm{P}_4^3\\mathrm{P}_{10}^3\\mathrm{P}_{12}^3$}", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "选择题", "ans": "", @@ -231188,7 +231913,8 @@ "content": "$100$件产品中有$97$件合格品与$3$件次品, 从中任意抽取$7$件进行检查;\\\\\n(1) 抽出的$7$件都是合格品的抽法有多少种?\\\\\n(2) 抽出的$7$件恰好有$2$件是次品的抽法有多少种?\\\\\n(3) 抽出的$7$件至少有$2$件是次品的抽法有多少种?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "解答题", "ans": "", @@ -231209,7 +231935,8 @@ "content": "求$\\mathrm{C}_{10}^1+2\\mathrm{C}_{10}^2+4\\mathrm{C}_{10}^3+\\cdots +2^9\\mathrm{C}_{10}^{10}$的值.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "二项式定理" ], "genre": "解答题", "ans": "", @@ -231252,7 +231979,8 @@ "content": "判断下列现象哪些是随机现象, 哪些不是随机现象;\\\\\n(1) 月球绕着地球转, 地球绕着太阳转;\\\\\n(2) 气压低的地方, 水的沸点低;\\\\\n(3) 黄浦江水位超出警戒线$1$米.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -231275,7 +232003,8 @@ "K0803002B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -231298,7 +232027,8 @@ "K0803002B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -231319,7 +232049,8 @@ "content": "已知某班有$38$名学生, 小李、小王、小张是该班的$3$名学生, 某次班会决定随机地挑选这$3$名学生在会上发言, 求下列事件出现的概率;\\\\\n(1) 小李、小王、小张按此次序被选中;\\\\\n(2) 小李、小王、小张按任意次序被选中.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -231342,7 +232073,8 @@ "K0803002B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "$\\dfrac 1{28}$, $\\dfrac 37$", @@ -231370,7 +232102,8 @@ "content": "一部$4$卷的文集, 按任意次序放到书架上, 求各卷自左向右或自右向左的卷号为$1$、$2$、$3$、$4$的概率.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -231393,7 +232126,8 @@ "content": "已知$10$个产品中有$3$个次品, 从中任取$5$个, 求至少有一个次品的概率.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -231416,7 +232150,8 @@ "content": "某种密码由$8$个数字组成, 且每个数字可以是$0$、$1$、$2$、…、$9$中的任意一个数, 求这种密码由完全不同的数字组成的概率.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -231437,7 +232172,8 @@ "content": "一工厂生产的$10$个产品中有$9$个一等品、$1$个二等品, 现从这批产品中抽取$4$个, 求其中恰好有一个二等品的概率.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -231462,7 +232198,8 @@ "K0803002B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -231483,7 +232220,8 @@ "content": "某城镇共有$10000$辆自行车, 牌照编号从$00001$到$10000$, 求在此城镇中偶然遇到的一辆自行车, 其牌照号码中有数字$8$概率.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -231508,7 +232246,8 @@ "K0807002B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -231531,7 +232270,8 @@ "K0807003B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "$0.97\\dot{3}$", @@ -231579,7 +232319,8 @@ "content": "将$n$间房间分给$n$个人, 每个人都以相等的可能性进入每一间房间. 而且每间房间里的人数没有限制, 求不出现空房的概率.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -231600,7 +232341,8 @@ "content": "把$10$本书随机地排在书架上, 求其中指定的$3$本书排在一起的概率.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -231621,7 +232363,8 @@ "content": "某人有$5$把钥匙, 但只有一把能打开门, 他每次取一把钥匙尝试开门, 求试到第$3$把钥匙时才打开门的概率.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -231644,7 +232387,8 @@ "content": "某次测验有$10$道备用试题, 甲同学在这$10$道题中能够答对$6$题, 现在备用试题中随机抽考$5$题, 规定答对$4$题或$5$题为优秀, 答对$3$题为及格;\\\\\n(1) 求甲同学获优秀的概率;\\\\\n(2) 求甲同学至少能够及格的概率.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -231665,7 +232409,8 @@ "content": "某中学有十八个班级, 每班选出三个代表出席学生代表会议, 从$54$名代表中任选$18$名组成工作委员会, 分别求下列事件的概率:\\\\\n(1) 高一(1)班在工作委员会中有代表;\\\\\n(2) 每个班级在工作委员会中都有代表.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -231688,7 +232433,8 @@ "K0803002B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -231709,7 +232455,8 @@ "content": "一批零件中有$9$个合格品和$3$个废品, 安装机器时, 从这批零件中随机取出一个, 如果每次取出的成品不放回去, 分别求在取得第$1$件合格品以前已取出$x$件废品数的概率, $x=0,1,2,3$.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -231732,7 +232479,8 @@ "K0806004B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -232263,7 +233011,8 @@ "content": "利用随机投点法求抛物线$y=x^2-4$与$x$轴组成的封闭图形的面积.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -232530,7 +233279,8 @@ "content": "有$4$名同学选报铅球、跳高、跳远三个体育项目, 如果每三个报一项, 那么共有多少种报名方法?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "加法原理与乘法原理" ], "genre": "解答题", "ans": "", @@ -232551,7 +233301,8 @@ "content": "从$6$种菜品种中选出$3$种, 分别种植在不同的$3$块地上进行试验, 有多少种不同的种植方法?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "解答题", "ans": "", @@ -232572,7 +233323,8 @@ "content": "已知$\\dfrac 1{\\mathrm{C}_5^n}-\\dfrac 1{\\mathrm{C}_6^n}=\\dfrac 7{10\\mathrm{C}_7^n}$, $n\\in \\mathbf{N}^*$, 求$\\mathrm{C}_8^n$;\\\\", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "解答题", "ans": "", @@ -232593,7 +233345,8 @@ "content": "已知$\\mathrm{P}_m^2=7\\mathrm{P}_{m-4}^2$, $m\\in \\mathbf{N}^*$, 求$m$的值.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "解答题", "ans": "", @@ -232638,7 +233391,8 @@ "content": "设$n\\in \\mathbf{N}^*$, 求证: $\\mathrm{C}_n^1+\\mathrm{C}_n^2+\\cdots +\\mathrm{C}_n^n=1+2+2^2+\\cdots +2^{n-1}$.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "二项式定理" ], "genre": "解答题", "ans": "", @@ -232661,7 +233415,8 @@ "K0803002B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -232684,7 +233439,8 @@ "K0803002B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -232705,7 +233461,8 @@ "content": "求$(1.009)^5$的近似值\\blank{50}(结果精确到$0.001$).", "objs": [], "tags": [ - "第八单元" + "第八单元", + "二项式定理" ], "genre": "填空题", "ans": "", @@ -232749,7 +233506,8 @@ "content": "$4$名男生、$4$名女生站站成一排, 男女间隔排列, 则不同的排法有\\blank{50}种.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "填空题", "ans": "", @@ -232856,7 +233614,8 @@ "content": "求$\\mathrm{C}_{3n}^{38-n}+\\mathrm{C}_{21+n}^{3n}$($n\\in \\mathbf{N}^*$)的值.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "解答题", "ans": "", @@ -232902,7 +233661,8 @@ "K0618001B" ], "tags": [ - "第八单元" + "第八单元", + "加法原理与乘法原理" ], "genre": "解答题", "ans": "", @@ -232967,7 +233727,8 @@ "content": "从$3$名男生和$n$名女生中, 任意选$3$人参加会议, 已知选出的$3$人中至少有一名女生的概率是$\\dfrac{34}{35}$, 求$n$的值.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -233009,7 +233770,8 @@ "content": "用随机投点法, 求$y=\\sin x(0\\le x\\le \\pi) $与$x$轴组成的封闭图形的面积.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -240065,7 +240827,8 @@ "K0801001B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -240090,7 +240853,8 @@ "K0801001B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -240114,7 +240878,8 @@ "K0802006B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -240138,7 +240903,8 @@ "K0802004B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "$\\Omega = \\{1,2,3,4,5,6\\}$, $A=\\{2,3,4,5,6\\}$, $B=\\{1,3,5\\}$, $C=\\{3,4,5,6\\}$", @@ -240169,7 +240935,8 @@ "K0802004B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -240192,7 +240959,8 @@ "K0803002B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -240216,7 +240984,8 @@ "K0806004B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -240239,7 +241008,8 @@ "K0803002B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -240263,7 +241033,8 @@ "K0806004B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "$P(A_0)=\\dfrac 13$, $P(A_1)=\\dfrac 12$, $P(A_2)=0$, $P(A_3)=\\dfrac 16$", @@ -240291,7 +241062,8 @@ "content": "写出例$6$中事件$A$、$B$各自包含的基本事件, 表示出$A\\cup B$与$A\\cap B$来验证例$6$中的结果.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -240314,7 +241086,8 @@ "K0805002B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -240337,7 +241110,8 @@ "K0806005B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "证明略", @@ -240367,7 +241141,8 @@ "K0806004B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "证明略", @@ -240397,7 +241172,8 @@ "K0806004B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -240420,7 +241196,8 @@ "K0807003B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -240443,7 +241220,8 @@ "K0807003B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -240466,7 +241244,8 @@ "K0807003B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "约$6.7\\times 10^4$条", @@ -240500,7 +241279,8 @@ "K0808002B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -240525,7 +241305,8 @@ "K0808003B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "$0.12$", @@ -240554,7 +241335,8 @@ "K0809001B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "(1) 证明略; (2) 证明略", @@ -240590,7 +241372,8 @@ "K0809002B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "(1) $pq$; (2) $1-p-q+pq$; (3) $p+q-pq$; (4) $1-pq$", @@ -240626,7 +241409,8 @@ "K0809003B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -240650,7 +241434,8 @@ "content": "为了解上海市某区居民用户的月平均用水量, 通过简单随机抽样获取了$100$户居民用户的月平均用水量. 在这个问题中, 总体和样本分别是什么?", "objs": [], "tags": [ - "第九单元" + "第九单元", + "概率" ], "genre": "解答题", "ans": "", @@ -244828,7 +245613,8 @@ "content": "公园有$4$个门, 从一个门进, 再从另一个门出, 共有多少种不同的走法?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "加法原理与乘法原理" ], "genre": "解答题", "ans": "", @@ -244849,7 +245635,8 @@ "content": "$4$名学生报名参加两项体育比赛, 每名学生可参加的比赛数目不限, 每项比赛参加的人数不限, 共有多少种不同的报名结果?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "加法原理与乘法原理" ], "genre": "解答题", "ans": "", @@ -244870,7 +245657,8 @@ "content": "在平面直角坐标系中, 以$1$、$2$、$3$、$4$、$5$这五个数中的两个分别作为一个点的横坐标和纵坐标, 可以组成多少个位于直线$y=x$下方的点?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "加法原理与乘法原理" ], "genre": "解答题", "ans": "", @@ -244891,7 +245679,8 @@ "content": "书架上放有$6$本不同的数学书和$5$本不同的语文书. 从中任取一本, 有多少种不同的取法?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "加法原理与乘法原理" ], "genre": "解答题", "ans": "", @@ -244912,7 +245701,8 @@ "content": "写出从$a$、$b$、$c$、$d$、$e$这五个不同元素中任意取出两个元素的所有排列.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "解答题", "ans": "", @@ -244933,7 +245723,8 @@ "content": "已知$M=\\{1,2,3,4\\}$, 且$m\\in M$, $n\\in M$, 方程$\\dfrac{x^2}m+\\dfrac{y^2}n=1$表示的曲线是椭圆. 问: 可以有多少个不同的椭圆?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "加法原理与乘法原理" ], "genre": "解答题", "ans": "", @@ -244954,7 +245745,8 @@ "content": "$5$名篮球队员甲、乙、丙、丁、戊, 排成一排.\\\\\n(1) 共有多少种不同的排法?\\\\\n(2) 若甲必须站在排头, 有多少种不同的排法?\\\\\n(3) 若甲不能站排头, 也不能站排尾, 有多少种不同的排法?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "解答题", "ans": "", @@ -244975,7 +245767,8 @@ "content": "(1) 配制某种染色剂, 需要加入$3$种有机染料、$2$种无机染料和$2$种添加剂, 其中有机染料的添加顺序不可以相邻. 为研究所有不同的添加顺序对染色效果的影响, 总共要试验多少次?\\\\\n(2) 某展览馆计划展出$10$幅不同的画, 其中水彩画$1$幅、油画$4$幅、国画$5$幅. 现排成一排陈列, 要求同一品种的画必须连在一起, 并且水彩画不放在两端. 问: 有多少种不同的陈列方式?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "解答题", "ans": "", @@ -244996,7 +245789,8 @@ "content": "已知$n$是正整数, 且$\\dfrac{\\mathrm{P}_n^7-\\mathrm{P}_n^5}{\\mathrm{P}_n^5} =89$. 求$n$的值.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "解答题", "ans": "", @@ -245017,7 +245811,8 @@ "content": "已知$n$为不小于$2$的正整数, 求证: $\\mathrm{P}_{n+1}^{n+1}-\\mathrm{P}_n^n=n^2\\mathrm{P}_{n-1}^{\nn-1}$.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "解答题", "ans": "", @@ -245038,7 +245833,9 @@ "content": "(1) 写出从$a$、$b$、$c$、$d$、$e$五个元素中任取两个不同元素的所有组合;\\\\\n(2) 写出从$a$、$b$、$c$、$d$、$e$五个元素中任取两个不同元素的所有排列.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合", + "排列" ], "genre": "解答题", "ans": "", @@ -245059,7 +245856,8 @@ "content": "平面上的$6$个点$A$、$B$、$C$、$D$、$E$、$F$中的任意$3$个点都不在同一条直线上, 写出所有以其中$3$个点为顶点的三角形.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "解答题", "ans": "", @@ -245080,7 +245878,8 @@ "content": "某班有$20$名男生、$18$名女生, 现从中任选$5$人组成一个宣传小组, 其中男生和女生都有的选法有多少种?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "解答题", "ans": "", @@ -245101,7 +245900,9 @@ "content": "从$1$、$2$、$3$、$4$、$5$这五个数字中任取两个不同的奇数和两个不同的偶数.\\\\\n(1) 一共有多少种不同的选法?\n(2) 可以组成多少个没有重复数字的四位奇数?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合", + "排列" ], "genre": "解答题", "ans": "", @@ -245122,7 +245923,8 @@ "content": "解关于正整数$x$的方程:\\\\\n(1) $\\mathrm{C}_{16}^{x^2-x}=\\mathrm{C}_{16}^{5x-5}$;\\\\\n(2) $\\mathrm{C}_{x+2}^{x-2}+\\mathrm{C}_{x+2}^{x-3}=\\dfrac 14\\mathrm{P}_{x+3}^3$.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "解答题", "ans": "", @@ -245143,7 +245945,8 @@ "content": "观察下列等式及其所示的规律:\\\\\n\\begin{align*}\n\\mathrm{C}_3^0+\\mathrm{C}_4^1=&\\mathrm{C}_4^0+\\mathrm{C}_4^1=\\mathrm{C}_5^1,\\\\\n\\mathrm{C}_3^0+\\mathrm{C}_4^1+\\mathrm{C}_5^2=&\\mathrm{C}_5^1+\\mathrm{C}_5^2=\\mathrm{C}_6^2,\\\\\n\\mathrm{C}_3^0+\\mathrm{C}_4^1+\\mathrm{C}_5^2+\\mathrm{C}_6^3=&\\mathrm{C}_6^2+\\mathrm{C}_6^3=\\mathrm{C}_7^3.\n\\end{align*}\n并据此化简$\\mathrm{C}_3^0+\\mathrm{C}_4^1+\\mathrm{C}_5^2+\\mathrm{C}_6^3+\\cdots+\\mathrm{C}_{n+3}^n$, 其中$n$为正整数.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "解答题", "ans": "", @@ -245164,7 +245967,8 @@ "content": "袋中装有$4$个红球、$3$个黄球、$3$个白球, 所有小球的大小与质地完全相同. 从中同时任取$2$个小球, 求取出的$2$个球颜色相同的概率.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -245185,7 +245989,8 @@ "content": "某校要从$2$名男生和$4$名女生中任选$4$人担任一项赛事的志愿者工作, 每个人被选中的可能性相同. 求在选出的志愿者中, 男生和女生都有的概率.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -245250,7 +246055,8 @@ "content": "(1) 若$(1-x)^6=a_0+a_1x+a_2x^2+\\cdots+a_6x^6$, 求$a_0+a_1+a_2+\\cdots+a_6$的值;\\\\\n(2) 已知$(x+1)^n=a_0+a_1(x-1)+a_2(x-1)^2+a_3(x-1)^3+\\cdots+a_n(x-1)^n$($n\\ge 2$, $n$为正整数), 求$a_0+a_1+a_2+\\cdots+a_n$的值.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "二项式定理" ], "genre": "解答题", "ans": "", @@ -245293,7 +246099,8 @@ "content": "一个家庭有两个孩子.\n(1) 已知年龄大的是女孩, 求年龄小的也是女孩的概率;\n(2) 已知其中一个是女孩, 求另一个也是女孩的概率.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -245314,7 +246121,8 @@ "content": "掷一颗骰子, 令事件$A=\\{2,3,5\\}$, $B=\\{1,2,4,5,6\\}$. 求$P(A)$、$P(B)$、$P(A\\cap B)$及$P(A|B)$.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -245335,7 +246143,8 @@ "content": "在一个盒子中有大小与质地相同的$20$个球, 其中$10$个红球, $10$个白球. 两人依次不放回地各摸$1$个球, 求:\n(1) 在第一个人摸出$1$个红球的条件下, 第二个人摸出$1$个白球的概率;\\\\\n(2) 第一个人摸出$1$个红球, 且第二个人摸出$1$个白球的概率.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -245356,7 +246165,8 @@ "content": "公司库房中的某个零件的$70\\%$来自$A$公司, $30\\%$来自$B$公司, 两个公司的正品率分别是$95\\%$和$90\\%$. 从库房中任取一个零件, 求它是正品的概率.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -245377,7 +246187,8 @@ "content": "盒子中有大小与质地相同的$5$个红球和$4$个白球, 从中随机取$1$个球, 观察其颜色后放回, 并同时放入与其相同颜色的球$3$个, 再从盒子中取$1$个球. 求第二次取出的球是白色的概率.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -245398,7 +246209,8 @@ "content": "从一个放有大小与质地相同的$3$个黑球、$2$个白球的袋子里摸出$2$个球并放入另外一个空袋子里, 再从后一个袋子里摸出$1$个球. 求该球是黑色的概率.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -245419,7 +246231,8 @@ "content": "设某公路上经过的货车与客车的数量之比为$2: 1$, 货车中途停车修理的概率为$0.02$, 客车中途停车修理的概率为$0.01$. 今有一辆汽车中途停车修理, 求该汽车是货车的概率.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -245440,7 +246253,8 @@ "content": "已知在所有男子中有$5\\%$患有色盲症, 在所有女子中有$0.25\\%$患有色盲症. 现随机抽取一人发现患有色盲症, 问: 其为男子的概率是多少? (设男子和女子的人数相等)", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -245461,7 +246275,8 @@ "content": "掷两颗骰子, 用$X$表示两点数差的绝对值. 求$X$的分布.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -245482,7 +246297,8 @@ "content": "以下是分布的为\\bracket{20}.\n\\fourch{$\\begin{pmatrix}0 & 1 \\\\ 1 & 1\\end{pmatrix}$}{$\\begin{pmatrix}-1 & 0 & 1\\\\ \\dfrac 12 & \\dfrac 13 & \\dfrac 16\\end{pmatrix}$}{$\\begin{pmatrix}1 & 2 & 3\\\\ \\dfrac 12 & \\dfrac 14 & \\dfrac 18\\end{pmatrix}$}{$\\begin{pmatrix}1 & 1.2 & 2 & 2.4 \\\\ -0.5 & 0.5 & 0.3 & 0.7\\end{pmatrix}$}", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "选择题", "ans": "", @@ -245503,7 +246319,8 @@ "content": "抛掷$4$枚硬币, 用$X$表示正面朝上的枚数. 求$X$的期望.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -245524,7 +246341,8 @@ "content": "从一个放有大小与质地相同的$5$个白球、$4$个黑球的罐子中不放回地摸$3$个球, 用$X$表示摸到的白球数. 求$X$的期望.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -245545,7 +246363,8 @@ "content": "设$X$是一个随机变量, $c$是常数. 求证: $X+c$的方差与$X$的方差相等.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -245566,7 +246385,8 @@ "content": "已知随机变量$X$的分布为$\\begin{pmatrix}1 & 2 & 3 \\\\ 0.4 & 0.2 & 0.4\\end{pmatrix}$, 求$X$的方差.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -245587,7 +246407,8 @@ "content": "已知随机变量$X$服从二项分布$B(n,p)$, 若$E[X]=30$, $D[X]=20$, 求$p$的值.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -245608,7 +246429,8 @@ "content": "一批产品的二等品率为$0.3$. 从这批产品中每次随机取一件, 并有放回地抽取$20$次. 用$X$表示抽到二等品的件数, 求$D[X]$.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -245631,7 +246453,8 @@ "content": "盒子中有大小与质地相同的$3$个白球、$1$个黑球, 若从中随机地摸出$2$个球, 求它们颜色不同的概率.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -245652,7 +246475,8 @@ "content": "从放有$6$黑$2$白共$8$颗珠子的袋子中抓$3$颗珠子, 分别求黑珠颗数$X$与白珠颗数$Y$的分布、期望与方差.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -245673,7 +246497,8 @@ "content": "从一副去掉大小王牌的$52$张扑克牌中任取$5$张牌, 求:\\\\\n(1) 至少有一张黑桃的概率;\\\\\n(2) 至少有一个对子(两张牌的数字一样)的概率.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -245694,7 +246519,8 @@ "content": "已知随机变量$X$服从正态分布$N(-2, \\sigma^2)$, 且$P(X\\le -1)=k$. 求$P(X\\le -3)$的值.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -245715,7 +246541,8 @@ "content": "某校高中三年级$1600$名学生参加了区第一次高考模拟统一考试, 已知数学考试成绩$X$服从正态分布$N(100, \\sigma^2)$(试卷满分为$150$分). 统计结果显示, 数学考试成绩在$80$分到$120$分之间的人数约为总人数的$\\dfrac 34$, 则此次统考中成绩不低于$120$分的学生人数约为\\bracket{20}.\n\\fourch{$80$}{$100$}{$120$}{$200$}", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "选择题", "ans": "", @@ -246217,7 +247044,8 @@ "content": "为了检查学生的身体素质指标, 从游泳类$1$项, 球类$3$项, 田径类$4$项共$8$项项目中随机抽取$4$项进行检测, 则每一类都被抽到的概率为\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "", @@ -246518,7 +247346,8 @@ "content": "从$1,2,3,4,5$中任取两个不同的数, 记``取到的两个数之和为偶数''为事件$A$, ``取到的两个数都大于$2$''为事件$B$, 则$P(A|B)=$\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "$\\dfrac 13$", @@ -246552,7 +247381,8 @@ "content": "若$P(A)=0.4$, $P(B)=0.6$, $P(B|A)=0.75$, 则$P(A|B)=$\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "$\\dfrac 12$", @@ -246586,7 +247416,8 @@ "content": "甲袋中有$3$个白球和$2$个红球, 乙袋中有$2$个白球和$3$个红球, 丙袋中有$4$个白球和$4$个红球. 先随机取一个袋子, 再从该袋中先后随机取$1$个球.则取出的球是红球的概率为\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "$\\dfrac 12$", @@ -246620,7 +247451,8 @@ "content": "在装有$6$个白球, $3$个黑球的袋子里随机摸$7$个球, 则摸出的白球个数$X$的期望为\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "$\\dfrac {14}3$", @@ -246654,7 +247486,8 @@ "content": "假设交通事故有$0.6$的概率是因为超速引起的, 则在$8$次交通事故中恰有$6$次是因为超速引起的概率为\\blank{50}(精确到$0.001$).", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "$0.209$", @@ -246688,7 +247521,8 @@ "content": "随机变量$X$的分布为\n\\[\\begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\\\ \\dfrac 18 & \\dfrac 14 & \\dfrac 14 & \\dfrac 14 & \\dfrac 18\\end{pmatrix},\\]\n则其期望$E[X]=$\\blank{50}; 方差$D[X]=$\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "$3$, $\\dfrac 32$", @@ -246722,7 +247556,8 @@ "content": "袋中有形状、大小完全相同的$5$个球, 编号分别为$1,2,3,4,5$. 从袋中取出$2$个球, 以$X$表示取出的$2$个球中的最大号码, 以$Y$表示取出的$2$个球中的最小号码. 则期望$E[(X+Y)(X-Y)]=$\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "$12$", @@ -246756,7 +247591,8 @@ "content": "对飞机进行射击, 按照受损伤影响的不同, 飞机的机身可分为两个部分. 要击落飞机, 必须在第一部分命中一次或在第二部分命中三次. 设炮弹击中飞机时, 命中第一部分的概率是$0.3$, 命中第二部分的概率是$0.7$, 射击进行到击落飞机为止. 则每次射击均命中的情况下, 击落飞机的命中次数的分布为\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "$\\begin{pmatrix} 1 & 2 & 3 \\\\ 0.3 & 0.21 & 0.49\\end{pmatrix}$", @@ -246790,7 +247626,8 @@ "content": "教室讲台上的文具盒里有$7$支红笔和$3$支黑笔, 它们的外观完全一样. 孔小姜有一个奇怪的习惯, 每当他随机地取用一支笔后, 如果这支笔是红笔, 那么他就把这支笔据为己有; 如果这支笔是黑笔, 那么他在归还这支笔的同时, 还额外往文具盒里多放一支黑笔(当然, 这些笔的外观还是一模一样, 无法分辨的). 当孔小姜第三次在文具盒里随机取用一支笔时, 他拿到黑笔的概率为\\blank{50}(精确到$0.001$).", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "$0.390$", @@ -246824,7 +247661,8 @@ "content": "生产方发出了一批产品, 产品共$50$箱, 其中误混了$2$箱不合格产品. 采购方接收该批产品的标准是: 从该批产品中任取$5$箱产品进行检测, 若至多有$1$箱不合格产品, 则接收该批产品. 问: 该批产品被接收的概率是多少?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "$\\dfrac{243}{245}$", @@ -246858,7 +247696,8 @@ "content": "飞机的几个发动机彼此独立工作, 测试表明某厂生产的每台发动机出现故障的概率均为$0.004$. 假设飞机正常飞行的条件是至少有一半的发动机能正常工作. 通过建模求解并回答: 一架搭载四台该厂生产的发动机的飞机与一台搭载两台该厂生产的发动机的飞机哪个更安全?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "搭载四台发动机的飞机更安全", @@ -246892,7 +247731,8 @@ "content": "设随机变量$X$的取值在集合$\\{0,1,2\\}$中.\\\\\n(1) 若$P(X=1)=\\dfrac 12$, 求期望$E[X]$的最大可能值$M$与$E[X]$的最小可能值$m$之差;\\\\\n(2) 猜测方差$D[X]$的最大可能值, 并证明你的猜测.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "$1$, $1$.", @@ -258794,7 +259634,8 @@ "K0802005B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -258817,7 +259658,8 @@ "K0802005B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "(1) 错误; (2) 错误; (3) 正确", @@ -258847,7 +259689,8 @@ "K0802006B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -258870,7 +259713,8 @@ "K0802005B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "选择题", "ans": "", @@ -258893,7 +259737,8 @@ "K0802005B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -258917,7 +259762,8 @@ "K0802004B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "$\\Omega=\\{W_1B_1,W_1B_2,W_1R_1,W_1R_2,W_1R_3,B_1B_2,B_1R_1,B_1R_2,B_1R_3,B_2R_1,B_2R_2,B_2R_3,R_1R_2,R_1R_3,R_2R_3\\}$, $A=\\{W_1B_1,W_1B_2,W_1R_1,W_1R_2,W_1R_3\\}$, $B=\\{W_1B_1,W_1B_2,B_1B_2,B_1R_1,B_1R_2,B_1R_3,B_2R_1,B_2R_2,B_2R_3\\}$. 或者$\\Omega' = \\{WB,WR,BB,BR,RR\\}$, $A'=\\{WB,WR\\}$, $B'=\\{WB,BB,BR\\}$", @@ -258948,7 +259794,8 @@ "K0802004B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "(1) $\\Omega = \\{BBGG,BGBG,BGGB,GGBB,GBGB,GBBG\\}$等; (2) $A=\\{BGBG,GBGB\\}$等; (3) $B=\\{BGBG,BGGB,GBGB,GBBG\\}$等", @@ -258978,7 +259825,8 @@ "K0803002B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -259001,7 +259849,8 @@ "K0803002B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "选择题", "ans": "", @@ -259024,7 +259873,8 @@ "K0805002B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -259047,7 +259897,8 @@ "K0805003B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "选择题", "ans": "B", @@ -259077,7 +259928,8 @@ "K0806004B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "$0.10$", @@ -259108,7 +259960,8 @@ "K0806004B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -259131,7 +259984,8 @@ "K0803002B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -259154,7 +260008,8 @@ "K0803002B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -259177,7 +260032,8 @@ "K0803002B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -259201,7 +260057,8 @@ "K0806004B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "$\\dfrac 35$", @@ -259231,7 +260088,8 @@ "K0806004B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -259254,7 +260112,8 @@ "K0807003B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -259278,7 +260137,8 @@ "K0807003B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "(1) $0.75$; (2) $15$", @@ -259313,7 +260173,8 @@ "K0807003B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "(1) $\\dfrac 79$; (2) 作``取两个球''的操作$n$次($n$充分大), 记录颜色不同的次数$S_n$, 计算$\\dfrac{S_n}n$", @@ -259342,7 +260203,8 @@ "K0808002B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "$0.75$", @@ -259371,7 +260233,8 @@ "K0803002B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -259396,7 +260259,8 @@ "K0808003B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "选择题", "ans": "C", @@ -259427,7 +260291,8 @@ "K0808003B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "(1) $\\dfrac 56$; (2) $\\dfrac 16$; (3) $\\dfrac 23$; (4) $\\dfrac 12$", @@ -259462,7 +260327,8 @@ "K0808002B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -259486,7 +260352,8 @@ "K0808002B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -266094,7 +266961,8 @@ "content": "$4$名学生分别报名参加学校的足球队、篮球队和棒球队, 每人限报其中的一支. 问: 有多少种不同的报名方法?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "加法原理与乘法原理" ], "genre": "解答题", "ans": "", @@ -266115,7 +266983,8 @@ "content": "某服装厂为学校设计了$4$种样式的上衣、$3$种样式的裤子. 若取其中的一件上衣和一条裤子配成校服, 则可以配出多少种不同样式的校服?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "加法原理与乘法原理" ], "genre": "解答题", "ans": "", @@ -266136,7 +267005,8 @@ "content": "在一种编码方式中, 每个编码都是两位字符, 规定第一位用数字$0$至$9$中之一, 第二位用$26$个小写英文字母中之一. 这种编码方式共可以产生多少个不同的编码?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "加法原理与乘法原理" ], "genre": "解答题", "ans": "", @@ -266157,7 +267027,8 @@ "content": "设集合$A=\\{(x, y)|x\\in \\mathbf{Z}, \\ y\\in \\mathbf{Z}, \\ \\text{且}|x|\\le 6, \\ |y|\\le 7\\}$, 则集合$A$中有多少个元素?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "加法原理与乘法原理" ], "genre": "解答题", "ans": "", @@ -266178,7 +267049,8 @@ "content": "从$a$、$b$、$c$、$d$、$e$这$5$个元素中取出$4$个, 放在$4$个不同的格子中, 且元素$b$不能放在第二个格子里. 问:一共有多少种不同的放法?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "解答题", "ans": "", @@ -266199,7 +267071,8 @@ "content": "$A$、$B$、$C$、$D$、$E$五人站成一排, 如果$B$必须站在$A$的右边($A$、$B$可以不相邻), 那么有多少种不同的排法?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "解答题", "ans": "", @@ -266242,7 +267115,8 @@ "content": "如图, 要接通从$A$到$B$的电路, 不同的接通方法有多少种?\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\draw (0,0) -- (0.5,0) (1.5,0) -- (2,0) (2,0.5) -- (2,-0.5) -- (2.5,-0.5) (2,0.5) -- (2.5,0.5) (3.5,0.5) -- (4,0.5) -- (4,-0.5) -- (3.5,-0.5) (4,0) -- (4.5,0) -- (4.5,2) -- (2.5,2) (1.5,2) -- (0,2) -- (0,0) (-0.5,1) -- (0,1) (4.5,1) -- (5,1);\n\\filldraw (0,1) circle (0.05) node [above left] {$A$} (4.5,1) circle (0.05) node [above right] {$B$};\n\\draw (0.5,0) --++ (15:1.1) (2.5,0.5) --++ (15:1.1) (2.5,-0.5) --++ (15:1.1) (1.5,2) --++ (15:1.1);\n\\filldraw [fill = white, draw = black] (0.5,0) circle (0.05) (2.5,0.5) circle (0.05) (2.5,-0.5) circle (0.05) (1.5,2) circle (0.05);\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第八单元" + "第八单元", + "加法原理与乘法原理" ], "genre": "解答题", "ans": "", @@ -266263,7 +267137,8 @@ "content": "用$1$、$2$、$3$、$4$、$5$、$6$组成没有重复数字的六位数, 要求所有相邻两个数字的奇偶性都不同, 且$1$和$2$相邻. 问:有多少个这样的六位数?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "解答题", "ans": "", @@ -266284,7 +267159,8 @@ "content": "已知$p_1$、$p_2$、$p_3$是互不相同的素数, $\\alpha$、$\\beta$、$\\gamma$是正整数, $n=p_1^\\alpha p_2^\\beta p_3^\\gamma$. 问: $n$有多少个不同的正约数?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "加法原理与乘法原理" ], "genre": "解答题", "ans": "", @@ -266305,7 +267181,8 @@ "content": "用$1$、$2$、$3$、$4$可以组成多少个没有重复数字的四位正整数? 其中有多少个偶数?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "解答题", "ans": "", @@ -266326,7 +267203,8 @@ "content": "从$1$、$2$、$3$、$4$、$5$这$5$个数字中, 任取$2$个不同的数字作为一个点的坐标, 一共可以组成多少个不同的点?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "解答题", "ans": "", @@ -266347,7 +267225,8 @@ "content": "在方程$ax+by=0$中, 设系数$a$、$b$是集合$\\{0, 1, 2, 3, 5, 7\\}$中两个不同的元素. 求这些方程所表示的不同直线的条数.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "加法原理与乘法原理" ], "genre": "解答题", "ans": "", @@ -266368,7 +267247,8 @@ "content": "将$5$个人排成一排, 若甲和乙必须排在一起, 则有多少种不同的排法?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "解答题", "ans": "", @@ -266389,7 +267269,8 @@ "content": "从$7$名运动员中选$4$名组成接力队参加$4\\times 100$米接力赛. 问:甲、乙两人都不跑中间两棒的排法有多少种?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "解答题", "ans": "", @@ -266410,7 +267291,8 @@ "content": "从$7$名男生和$5$名女生中选取$3$人依次进行面试.\\\\\n(1) 若参加面试的人全是女生, 则有多少种不同的面试方法?\n(2) 若参加面试的人中, 恰好有$1$名女生, 则有多少种不同的面试方法?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "解答题", "ans": "", @@ -266431,7 +267313,8 @@ "content": "若$m$为正整数, 且$m<27$, 则$(27-m)(28-m)\\cdots(34-m)$等于\\bracket{20}.\n\\fourch{$\\mathrm{P}_{27-m}^8$}{$\\mathrm{P}_{34-m}^{27-m}$}{$\\mathrm{P}_{34-m}^7$}{$\\mathrm{P}_{34-m}^8$}", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "选择题", "ans": "", @@ -266452,7 +267335,8 @@ "content": "求满足等式$\\mathrm{P}_{2n}^3=28\\mathrm{P}_n^2$的正整数$n$的值.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "解答题", "ans": "", @@ -266473,7 +267357,8 @@ "content": "解关于正整数$x$的不等式$\\mathrm{P}_8^x<6\\mathrm{P}_8^{x-2}$.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "解答题", "ans": "", @@ -266494,7 +267379,8 @@ "content": "有$4$张分别标有数字$1$、$2$、$3$、$4$的红色卡片和$4$张分别标有数字$1$、$2$、$3$、$4$的蓝色卡片, 从这$8$张卡片中取出$4$张排成一行. 如果所取出的$4$张卡片所标数字之和等于$10$, 那么不同的排法共有多少种?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "解答题", "ans": "", @@ -266515,7 +267401,8 @@ "content": "有$6$张连号的电影票, 分给$3$名教师和$3$名学生, 要求师生相间而坐. 求不同分法的种数.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "解答题", "ans": "", @@ -266536,7 +267423,8 @@ "content": "在一张节目单中原有$6$个节目已排好顺序, 现要插入$3$个节目, 并要求不改变原有$6$个节目前后相对顺序. 问: 一共有多少种不同的插法?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "解答题", "ans": "", @@ -266557,7 +267445,8 @@ "content": "$2$名男生和$4$名女生排成一排. 问: 男生既不相邻也不排两端的不同排法共有多少种?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "解答题", "ans": "", @@ -266578,7 +267467,8 @@ "content": "在一次电影展中, 某影院要在两天内放映$12$部参赛影片, 每天只有$6$个时间段放映$6$部参赛影片, 每个时间段放映$1$部, 其中甲、乙两部电影不能在同一天放映. 问: 有多少种不同的排片方案?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "解答题", "ans": "", @@ -266599,7 +267489,8 @@ "content": "从$6$人中选取$4$人分别去$A$、$B$、$C$、$D$四个城市游览, 要求每个城市有一人游览, 而每人只游览一个城市, 且这$6$人中, 甲、乙两人都不去$A$地游览. 问: 不同的选择方案共有多少种?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "解答题", "ans": "", @@ -266620,7 +267511,8 @@ "content": "平面上有$10$个点, 其中有$4$个点在同一条直线上, 除此以外, 不再有三点共线. 问: 由这些点可以确定多少条直线?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "解答题", "ans": "", @@ -266641,7 +267533,9 @@ "content": "(1) 从$10$男$8$女中任选$5$人, 共有多少种不同的选法?\n(2) 从$10$男$8$女中任选$5$人(男女都有)担任$5$项不同的工作, 共有多少种不同的分配方法?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合", + "排列" ], "genre": "解答题", "ans": "", @@ -266662,7 +267556,8 @@ "content": "从$5$名男生和$3$名女生中各任选$2$名参加一个歌唱小组, 有多少种不同的选择方案?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "解答题", "ans": "", @@ -266683,7 +267578,8 @@ "content": "某批次$200$件产品中有$5$件次品, 现从该批次中任取$4$件产品.\\\\\n(1) 若$4$件产品都不是次品, 则这样的取法有多少种?\\\\\n(2) 若$4$件产品中至少有$1$件次品, 则这样的取法有多少种?\\\\\n(3) 若$4$件产品不都是次品, 则这样的取法有多少种?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "解答题", "ans": "", @@ -266704,7 +267600,8 @@ "content": "从$5$名外语系大学生中任选$4$名参加翻译、交通、礼仪三项义工活动, 要求翻译有$2$人参加, 交通和礼仪各有$1$人参加. 问: 不同的分配方法共有多少种?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "解答题", "ans": "", @@ -266725,7 +267622,8 @@ "content": "某小组共有$10$名学生, 其中女生$3$名. 现任选$2$名代表, 则至少有$1$名女生当选的选法有多少种?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "解答题", "ans": "", @@ -266746,7 +267644,8 @@ "content": "设$n$为正整数, 求值:\\\\\n(1) $\\mathrm{C}_{2n-3}^{n-1}+\\mathrm{C}_{n+1}^{2n-3}$;\\\\\n(2) $\\mathrm{C}_{13+n}^{3n}+\\mathrm{C}_{12+n}^{3n-1}+\\mathrm{C}_{11+n}^{3n-2}+\\cdots+\\mathrm{C}_{2n}^{17-n}$.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "解答题", "ans": "", @@ -266767,7 +267666,8 @@ "content": "求满足等式$\\mathrm{C}_{18}^k=\\mathrm{C}_{18}^{2k-3}$的所有正整数$k$.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "解答题", "ans": "", @@ -266788,7 +267688,8 @@ "content": "证明: $\\mathrm{C}_n^m=\\dfrac{m+1}{n+1}\\mathrm{C}_{n+1}^{m+1}$, 其中$m$是自然数, $n$是正整数, 且$m\\le n$.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "解答题", "ans": "", @@ -266809,7 +267710,8 @@ "content": "把$4$本不同的书全部分给$3$名学生, 每人至少$1$本, 有多少种不同的分法?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "解答题", "ans": "", @@ -266830,7 +267732,8 @@ "content": "袋中装有$m$个红球和$n$个白球, 且$m\\ge n\\ge 2$. 这些红球和白球的大小及质地都相同. 从袋中同时任取$2$个球, 若$2$个球都是红球的取法总数是$2$个球颜色不同的取法总数的整数倍, 求证: $m$必为奇数.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "解答题", "ans": "", @@ -266851,7 +267754,8 @@ "content": "如图, 在$\\angle AOB$的两边$OA$、$OB$上分别有$5$个点和$6$个点(都不同于点$O$), 这连同点$O$在内的$12$个点可以确定多少个不同的三角形? \n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.7]\n\\draw (0,0) node [left] {$O$} coordinate (O);\n\\draw (5,0) node [right] {$B$} coordinate (B);\n\\draw (45:5) node [right] {$A$} coordinate (A);\n\\foreach \\i in {0,0.7,...,4.3} {\\filldraw (\\i,0) circle (0.03);};\n\\foreach \\i in {0,0.8,...,4.2} {\\filldraw (45:\\i) circle (0.03);};\n\\draw (O) -- (A) (O) -- (B);\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "解答题", "ans": "", @@ -266872,7 +267776,8 @@ "content": "有$12$名翻译人员, 其中$3$人只能翻译英语, $4$人只能翻译法语, 其余$5$人既能翻译英语, 也能翻译法语. 从这$12$名翻译人员中任选$6$人, 其中$3$人翻译英语, $3$人翻译法语, 有多少种不同的分配方法?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "解答题", "ans": "", @@ -266893,7 +267798,8 @@ "content": "利用组合数的性质化简: $\\mathrm{C}_3^3+\\mathrm{C}_4^3+\\mathrm{C}_5^3+\\cdots+\\mathrm{C}_n^3$.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "解答题", "ans": "", @@ -266916,7 +267822,8 @@ "content": "将两颗质地均匀的骰子同时抛掷一次, 求向上的点数之和为$5$的概率.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -266937,7 +267844,8 @@ "content": "用$1$、$2$、$3$、$4$、$5$组成没有重复数字的三位数, 从中随机地取一个, 求取到的数为奇数的概率.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -266958,7 +267866,8 @@ "content": "从甲、乙、丙、丁、戊五人中任选两人参加一项活动, 求甲、乙两人中至少有一人被选中的概率.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -266979,7 +267888,8 @@ "content": "在$10$件产品中有$8$件一等品、$2$件二等品, 从中随机抽取$2$件产品. 求取到的产品中至多有$1$件二等品的概率.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -267000,7 +267910,8 @@ "content": "某校高一年级举行演讲比赛, 共有$10$名学生参赛, 其中一班有$3$名, 二班有$2$名, 其他班有$5$名. 若采用抽签的方式确定他们的演讲顺序, 求一班的$3$名学生恰好被排在一起(指演讲序号相连)的概率.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -267131,7 +268042,8 @@ "content": "已知$(1+x)^{10}=a_0+a_1(1-x)+a_2(1-x)^2+\\cdots+a_{10}(1-x)^{10}$, 求$a_8$的值.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "二项式定理" ], "genre": "解答题", "ans": "", @@ -267218,7 +268130,8 @@ "content": "掷一颗骰子所得的样本空间为$\\Omega=\\{1, 2, 3, 4, 5, 6\\}$. 令事件$A=\\{2, 3, 5\\}$, $B=\\{1, 2, 4, 5, 6\\}$. 求$P(B|A)$.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -267239,7 +268152,8 @@ "content": "将一枚质地均匀的硬币抛掷$2$次, 设事件$A$为``第一次出现正面'', 事件$B$为``第二次出现正面''. 求$P(A|B)$与$P(B|A)$.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -267260,7 +268174,8 @@ "content": "某工厂有四条流水线生产同一产品, 已知这四条流水线的产量分别占总产量的$15\\%$、$20\\%$、$30\\%$和$35\\%$, 又知这四条流水线的产品不合格率依次为$0.05$、$0.04$、$0.03$和$0.02$. 从该厂的这一产品中任取一件, 抽到不合格品的概率是多少?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -267281,7 +268196,8 @@ "content": "假设有两箱同种零件, 第一箱内装有$50$件, 其中$10$件为一等品; 第二箱内装有$30$件, 其中$18$件为一等品(两箱外观相同). 现从两箱中随意挑出一箱, 然后从该箱中先后随机地取出两个零件(取出的零件不放回). 求先取出的零件是一等品的概率.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -267302,7 +268218,8 @@ "content": "设某种动物活到$20$岁的概率为$0.8$, 活到$25$岁的概率为$0.4$. 现有一只$20$岁的该种动物, 它活到$25$岁的概率是多少?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -267323,7 +268240,8 @@ "content": "袋中装有编号为$1$到$N$的$N$个球. 先从袋中任取一个球, 若该球不是$1$号球, 则放回袋中; 若是$1$号球, 则不放回, 然后再摸一次. 求第二次摸到$2$号球的概率.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -267344,7 +268262,8 @@ "content": "一袋中装有$6$个大小与质地相同的白球, 编号为$1$、$2$、$3$、$4$、$5$、$6$. 从该袋内随机取出$3$个球, 记被取出球的最大号码数为$X$. 写出随机变量$X$的分布.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -267365,7 +268284,8 @@ "content": "掷两颗骰子, 用$X$表示较大的点数(在点数相同时, $X$表示共同的点数). 求$X$的分布与期望.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -267386,7 +268306,8 @@ "content": "设某射手打靶环数$X$的分布为$\\begin{pmatrix} 7 & 8 & 9 & 10 \\\\ a & 0.1 & 0.3 & b \\end{pmatrix}$, 已知期望$E[X]=8.9$. 求$a$、$b$的值.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -267407,7 +268328,8 @@ "content": "一袋中装有大小与质地相同的$2$个白球和$3$个黑球.\\\\\n(1) 从中有放回地依次摸出$2$个球, 求$2$球颜色不同的概率;\\\\\n(2) 从中不放回地依次摸出$2$个球, 记$2$球中白球的个数为$X$. 求$X$的期望和方差.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -267428,7 +268350,8 @@ "content": "编号为$1$、$2$、$3$、$4$的四名学生随机入座编号为$1$、$2$、$3$、$4$的座位, 每个座位坐一人. 座位编号和学生编号一样时称为一个配对. 用$X$表示配对数, 求$E[X]$.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -267449,7 +268372,8 @@ "content": "已知一个随机变量$X$的分布为$\\begin{pmatrix}-1 & 0 & 1 \\\\ a & b & c \\end{pmatrix}$. 若$a+c=2b$, 且$E[X]=\\dfrac 13$, 求$D[X]$的值.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -267472,7 +268396,8 @@ "content": "一袋中装有编号为$1$、$2$、$3$、$4$、$5$的五个大小与质地相同的球. 依次摸两个球, 用$X_1$、$X_2$分别表示第一个及第二个球的编号. 在以下两种情况下分别求$X_1$、$X_2$以及两编号之和$X_1+X_2$的分布, 再分别验证等式$E[X_1+X_2]=E[X_1]+E[X_2]$与$D[X_1+X_2]=D[X_1]+D[X_2]$是否成立.\\\\\n(1) 放回;\\\\\n(2) 不放回.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -267493,7 +268418,8 @@ "content": "先掷一颗骰子, 记朝上的点数为$X$. 再抛掷$X$枚硬币, 记$Y$为正面朝上的硬币数. 求$Y$的分布、期望与方差.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -267514,7 +268440,8 @@ "content": "一名学生每天骑车上学, 从家到学校的途中经过$6$个路口. 假设他在各个路口遇到红灯的事件是相互独立的, 并且概率都是$\\dfrac 13$.\\\\\n(1) 用$X$表示这名学生在途中遇到红灯的次数, 求$X$的分布;\\\\\n(2) 求这名学生在途中至少遇到一次红灯的概率.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -267535,7 +268462,8 @@ "content": "从有$7$名男生的$15$名学生中任意选择$10$名, 用$X$表示其中的男生人数.求$P(X=4)$的值.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -267556,7 +268484,8 @@ "content": "某学生参加一次考试, 已知在备选的$10$道试题中, 能答对其中的$6$道题. 规定每次考试都从备选题中随机抽出$3$道题进行测试, 求该生答对试题数$X$的分布.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -267577,7 +268506,8 @@ "content": "从一副去掉大小王牌的$52$张扑克牌中任取$5$张牌, 用$X$表示其中黑桃的张数. 求$X$的分布、期望与方差.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -267598,7 +268528,8 @@ "content": "从装有大小与质地相同的$a$个白球、$b$个黑球的袋子中不放回地随机取$n$个球, $n$不能超过总个数$a+b$. 用$X$表示其中的白球个数. 这可以想象成依次取球, 用$X_k$表示第$k$次取球的结果: 如果是白球, $X_k=1$; 如果是黑球, $X_k=0$($k=1, 2, \\cdots, n$). 并设$X=X_1+X_2+\\cdots+X_n$, 表示取出的白球的总数. 设$n=2$. 求:\\\\\n(1) $E[X_1X_2]$;\\\\\n(2) $E[X^2]$与$D[X]$.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -267619,7 +268550,8 @@ "content": "已知随机变量$X$服从正态分布$N(3,\\sigma^2)$, 且$P(1\\le X\\le 5)=0.6$. 求$P(X>5)$的值.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "解答题", "ans": "", @@ -268246,7 +269178,8 @@ "content": "甲、乙两人从$5$门不同的选修课中各选修$2$门, 则甲、乙所选的课程中恰有$1$门相同的选法有\\blank{50}种.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "加法原理与乘法原理" ], "genre": "填空题", "ans": "$60$", @@ -268929,7 +269862,8 @@ "content": "一个袋中装有同样大小、质量的$10$个球, 其中$2$个红球、$3$个蓝球、$5$个黑球. 经过充分混合后, 若从此袋中任意取出$4$个球, 则三种颜色的球均取到的概率为\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "$\\dfrac 12$", @@ -269595,7 +270529,8 @@ "content": "某高级中学欲从本校的$7$位古诗词爱好者(其中男生$2$人、女生$5$人)中随机选取3名同学作为学校诗词朗读比赛的主持人, 若要求主持人中至少有一位是男同学, 则不同选取方法的种数是\\blank{50}(结果用数值表示).", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "填空题", "ans": "", @@ -270039,7 +270974,8 @@ "content": "新冠病毒爆发初期, 全国支援武汉的活动中, 需要从$A$医院某科室的$6$名男医生(含一名主任医师)、$4$名女医生(含一名主任医师)中分别选派$3$名男医生和$2$名女医生, 要求至少有一名主任医师参加, 则不同的选派方案共有\\blank{50}种. (用数字作答)", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "填空题", "ans": "", @@ -270085,7 +271021,8 @@ "content": "某校开设$9$门选修课程, 其中$A$, $B$, $C$三门课程由于上课时间相同, 至多选一门, 若规定每位学生选修$4$门, 则一共有\\blank{50}种不同的选修方案.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "填空题", "ans": "", @@ -270544,7 +271481,8 @@ "content": "疫情期间家长会, 我校要从$5$名男生, $3$名女生中选派$4$名志愿者担任家长入校测量体温、查看行程码、健康码、登记信息四项不同的工作, 若其中女生不能从事测量体温, 则不同的选派方案共有\\blank{50}种.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "填空题", "ans": "", @@ -271410,7 +272348,8 @@ "content": "从甲、乙、丙、丁$4$名同学中随机选$2$名同学参加志愿者服务, 则甲、乙两人都没有被选到的概率为\\blank{50}(用数字作答).", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "", @@ -272897,7 +273836,8 @@ "content": "若$\\mathrm{P}_n^2=6$, 则$n=$\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "填空题", "ans": "", @@ -272988,7 +273928,8 @@ "content": "若掷一颗质地均匀的骰子, 则出现向上的点数大于$4$的概率是\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "", @@ -273615,7 +274556,8 @@ "content": "在$100$件产品中有$90$件一等品, $10$件二等品, 从中随机取出$4$件产品.\n则恰含$1$件二等品的概率是\\blank{50}. (结果精确到$0.01$)", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "", @@ -275620,7 +276562,8 @@ "content": "两个三口之家, 共$4$个大人, $2$个小孩, 约定星期日乘红色、白色两辆轿车结伴郊游, 每辆车最多乘坐$4$人, 其中两个小孩不能独坐一辆车, 则不同的乘车方法种数是\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "填空题", "ans": "", @@ -276079,7 +277022,8 @@ "content": "从$m$($m\\in \\mathbf{N}^*$, $m\\ge 4$)个男生、$6$个女生中任选$2$个人当发言人, 假设事件$A$表示选出的$2$个人性别相同, 事件$B$表示选出的$2$个人性别不同. 如果$A$的概率和$B$的概率相等, 则$m=$\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "", @@ -276227,7 +277171,8 @@ "content": "设集合$S=\\{1,2,3,\\cdots,2020\\}$, 设集合$A$是集合$S$的非空子集, $A$中的最大元素和最小元素之差称为集合$A$的直径. 那么集合$S$所有直径为$71$的子集的元素个数之和为\\bracket{20}.\n\\fourch{$71\\cdot 1949$}{$2^{70}\\cdot 1949$}{$2^{70}\\cdot 37\\cdot 1949$}{$2^{70}\\cdot 72\\cdot 1949$}", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "选择题", "ans": "", @@ -276927,7 +277872,8 @@ "content": "在$4$名男生, $4$名女生中随机选出$3$名学生参加某次活动, 则选出的学生中至多$1$名女生的概率为\\blank{50}(结果用最简分数表示).", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "", @@ -277430,7 +278376,8 @@ "content": "甲、乙、丙、丁$4$名同学参加志愿者服务, 分别到三个路口疏导交通, 每个路口有$1$名或$2$名志愿者. 若每种分配方案的可能性相等, 则甲、乙两人在同一路口的概率为\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "", @@ -278279,7 +279226,8 @@ "content": "投掷两颗六个面上分别刻有$1$到$6$的点数的均匀的骰子, 得到其向上的点数分别为$m$和$n$, 则复数$\\dfrac{m+n\\mathrm{i}}{n+m\\mathrm{i}}$为虚数的概率为\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "", @@ -278344,7 +279292,8 @@ "K0615004B" ], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "", @@ -278840,7 +279789,8 @@ "content": "某商场举行购物抽奖促销活动, 规定每位顾客从装有编号为$0$、$1$、$2$、$3$的四个相同小球的抽奖箱中, 每次取出一球记下编号后放回, 连续取两次, 若取出的两个小球编号相加之和等于$6$, 则中一等奖, 等于$5$中二等奖, 等于$4$或$3$中三等奖. 则顾客抽奖中三等奖的概率为\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "", @@ -279316,7 +280266,8 @@ "content": "高三某位同学参加物理、化学、政治科目的等级考, 已知这位同学在物理、化学、政治科目考试中达$A^+$的概率分别为$\\dfrac 78$、$\\dfrac 34$、$\\dfrac 5{12}$, 这三门科目考试成绩的结果互不影响, 则这位考生至少得$2$个$A^+$的概率是\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "", @@ -279740,7 +280691,8 @@ "content": "新冠病毒爆发初期, 全国支援武汉的活动中, 需要从$A$医院某科室的$7$名男医生(含一名主任医师)、 $5$名女医生(含一名主任医师)中分别选派$3$名男医生和 $2$名女医生, 要求至少有一名主任医师参加, 则不同的选派方案共有\\blank{50}种.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "填空题", "ans": "", @@ -280249,7 +281201,8 @@ "content": "甲、乙、丙三个不同单位的医疗队里各有$3$人, 职业分别为医生、护士与化验师, 现在要从中抽取$3$人组建一支志愿者队伍, 则他们的单位与职业都不相同的概率是\\blank{50}(结果用最简分数表示).", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "", @@ -281174,7 +282127,8 @@ "content": "甲、乙两队进行排球决赛, 现在的情形是甲队只要再赢一局就获冠军, 乙队需要再赢两局才能得冠军.若两队在每局赢的概率都是$0.5$, 则甲队获得冠军的概率为\\blank{50}.(结果用数值表示)", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "", @@ -282088,7 +283042,8 @@ "content": "在报名的$8$名男生和$5$名女生中, 选取$6$人参加志愿者活动, 要求男、女都有, 则不同的选取方式的种数为\\blank{50}(结果用数值表示).", "objs": [], "tags": [ - "第八单元" + "第八单元", + "组合" ], "genre": "填空题", "ans": "", @@ -282135,7 +283090,8 @@ "content": "若$A$、$B$满足$P(A)=\\dfrac 12$, $P(B)=\\dfrac 45$, $P(AB)=\\dfrac 25$, 则$P(\\overline{A}B)-P(A\\overline{B})=$\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "", @@ -282911,7 +283867,8 @@ "content": "若排列数$\\mathrm{P}_6^m=6\\times 5\\times 4$, 则$m=$\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "排列" ], "genre": "填空题", "ans": "", @@ -283511,7 +284468,8 @@ "content": "随机变量$\\xi$的概率分布率由下表给出: \n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline\n$x$ & $7$ & $8$ & $9$ & $10$ \\\\ \\hline\n$P(\\xi=x)$ & $0.3$ & $0.35$ & $0.2$ & $0.15$ \\\\ \\hline\n\\end{tabular}\n\\end{center}\n则随机变量$\\xi$的均值是\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "", @@ -283574,7 +284532,8 @@ "content": "从一副混合后的扑克牌($52$张)中随机抽取$1$张, 事件$A$为``抽得红桃K'', 事件$B$为``抽得为黑桃'', 则概率$P(A\\cup B)=$\\blank{50}.(结果用最简分数表示)", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "", @@ -283685,7 +284644,8 @@ "content": "在集合$U=\\{a,b,c,d\\}$的子集中选出$2$个不同的子集, 需同时满足以下两个条件: \\textcircled{1} $a$、$b$都要选出; \\textcircled{2} 对选出的任意两个子集$A$和$B$, 必有$A\\subseteq B$或$B\\subseteq A$, 那么共有\\blank{50}种不同的选法.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "加法原理与乘法原理" ], "genre": "填空题", "ans": "", @@ -284081,7 +285041,8 @@ "content": "马老师从课本上抄录一个随机变量$\\xi$的概率分布律如下表请小牛同学计算$\\xi$的数学期望, 尽管``!''处无法完全看清, 且两个``?''处字迹模糊, 但能肯定这两个``?''处的数值相同. 据此, 小牛给出了正确答案$E\\xi=$\\blank{50}.\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|}\n\\hline\n$x$ & $1$ & $2$ & $3$ \\\\ \\hline\n$P(\\xi=x)$ & ? & ! & ? \\\\ \\hline\n\\end{tabular}\n\\end{center}", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "", @@ -284146,7 +285107,8 @@ "content": "随机抽取$9$个同学中, 至少有$2$个同学在同一月出生的概率是\\blank{50}(默认每月天数相同, 结果精确到$0.001$).", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "", @@ -284637,7 +285599,8 @@ "content": "三位同学参加跳高、跳远、铅球项目的比赛, 若每人都选择其中两个项目, 则有且仅有两人选择的项目完全相同的概率是\\blank{50}(结果用最简分数表示).", "objs": [], "tags": [ - "第八单元" + "第八单元", + "概率" ], "genre": "填空题", "ans": "", @@ -284764,7 +285727,8 @@ "content": "设$10\\le x_1D\\xi_2$}{$D\\xi_1=D\\xi_2$}{$D\\xi_1