From 56062ec9bcfca5ae3bd4d4be783f48947bb3077d Mon Sep 17 00:00:00 2001
From: wangweiye7840 <kj_wangweiye@126.com>
Date: Fri, 7 Jul 2023 14:48:45 +0800
Subject: [PATCH] =?UTF-8?q?=E5=BD=95=E5=85=A5=E7=A9=BA=E4=B8=AD=E8=AF=BE?=
 =?UTF-8?q?=E5=A0=82=E9=80=89=E6=8B=A9=E6=80=A7=E5=BF=85=E4=BF=AE=E7=AC=AC?=
 =?UTF-8?q?=E4=BA=8C=E5=86=8C=E8=AE=A1=E6=95=B0=E5=8E=9F=E7=90=86=E4=BE=8B?=
 =?UTF-8?q?=E9=A2=98=E4=B8=8E=E4=B9=A0=E9=A2=98=20=E5=B9=B6=E5=AF=B9?=
 =?UTF-8?q?=E5=BA=94=E5=8D=95=E5=85=83?=
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

---
 工具v2/批量收录题目.py |    2 +-
 题库0.3/Problems.json  | 1628 ++++++++++++++++++++++++++++++++++++++++
 2 files changed, 1629 insertions(+), 1 deletion(-)

diff --git a/工具v2/批量收录题目.py b/工具v2/批量收录题目.py
index c87f6bc3..67130a26 100644
--- a/工具v2/批量收录题目.py
+++ b/工具v2/批量收录题目.py
@@ -1,5 +1,5 @@
 #修改起始id,出处,文件名
-starting_id = 19054 #起始id设置, 来自"寻找空闲题号"功能
+starting_id = 19118 #起始id设置, 来自"寻找空闲题号"功能
 raworigin = "" #题目来源的前缀(中缀在.tex文件中)
 filename = r"C:\Users\wangweiye\Documents\wwy sync\临时工作区\空中课堂必修第二册例题与习题.tex" #题目的来源.tex文件
 editor = "王伟叶" #编辑者姓名
diff --git a/题库0.3/Problems.json b/题库0.3/Problems.json
index 812c03cb..a191d64f 100644
--- a/题库0.3/Problems.json
+++ b/题库0.3/Problems.json
@@ -490273,6 +490273,1634 @@
         "space": "4em",
         "unrelated": []
     },
+    "019118": {
+        "id": "019118",
+        "content": "一个三层的书架上共放有$9$本书, 其中第一层放有$4$本不同的语文书, 第二层放有$3$本不同的数学书, 第三层放有$2$本不同的外语书. 若从书架的第一、二、三层各取$1$本书, 共有多少种不同的取法?",
+        "objs": [],
+        "tags": [
+            "第八单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第二册计数原理例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "019119": {
+        "id": "019119",
+        "content": "用$1$、$2$、$3$、$4$、$5$可以组成多少个没有重复数字的三位数?",
+        "objs": [],
+        "tags": [
+            "第八单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第二册计数原理例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "019120": {
+        "id": "019120",
+        "content": "用$1$、$2$、$3$、$4$、$5$可以组成多少个没有重复数字的三位奇数?",
+        "objs": [],
+        "tags": [
+            "第八单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第二册计数原理例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "019121": {
+        "id": "019121",
+        "content": "正整数$540$有多少个不同的正约数?",
+        "objs": [],
+        "tags": [
+            "第八单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第二册计数原理例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "019122": {
+        "id": "019122",
+        "content": "将标有$1$、$2$、$3$、$4$的四个球放到标号为$1$、$2$、$3$、$4$的四只盒子里, 每只盒子放一个球, 则每一只盒子的标号与该盒子里的球的标号均不相同的放法共有种.",
+        "objs": [],
+        "tags": [
+            "第八单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第二册计数原理例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "019123": {
+        "id": "019123",
+        "content": "用$0$、$1$、$2$、$3$、$4$、$5$组成没有重复数字的六位数, 其中不小于$201345$的数共有多少个?",
+        "objs": [],
+        "tags": [
+            "第八单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第二册计数原理例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "019124": {
+        "id": "019124",
+        "content": "用$1$、$2$、$3$、$4$、$5$这五个数字可以组成多少个十位数字大于个位数字的两位数?",
+        "objs": [],
+        "tags": [
+            "第八单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第二册计数原理例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "019125": {
+        "id": "019125",
+        "content": "如图, 从甲地到乙地有$2$条路, 从乙地到丁地有$3$条路; 从甲地到丙地有$4$条路, 从丙地到丁地有$2$条路. 那么, 从甲地到丁地, 如果每条路至多走一次, 且每个地点至多经过一次, 共有多少种不同的走法?\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0.1,0) -- (0.1,2) (-0.1,0) -- (-0.1,2) (-0.3,0) -- (-0.3,2) (0.3,0) -- (0.3,2);\n\\draw (0,0.3) -- (2,0.3) (0,-0.3) -- (2,-0.3);\n\\draw (0,2.3) -- (2,2.3) (0,1.7) -- (2,1.7);\n\\draw (1.7,0) -- (1.7,2) (2,0) -- (2,2) (2.3,0) -- (2.3,2);\n\\filldraw [fill = white] (0,0) circle (0.4) node {丙};\n\\filldraw [fill = white] (0,2) circle (0.4) node {甲};\n\\filldraw [fill = white] (2,0) circle (0.4) node {丁};\n\\filldraw [fill = white] (2,2) circle (0.4) node {乙};\n\\end{tikzpicture}\n\\end{center}",
+        "objs": [],
+        "tags": [
+            "第八单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第二册计数原理例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "019126": {
+        "id": "019126",
+        "content": "在$300$和$800$之间, 共有多少个没有重复数字的奇数?",
+        "objs": [],
+        "tags": [
+            "第八单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第二册计数原理例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "019127": {
+        "id": "019127",
+        "content": "在$300$和$800$之间, 共有多少个没有重复数字的偶数?",
+        "objs": [],
+        "tags": [
+            "第八单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第二册计数原理例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "019128": {
+        "id": "019128",
+        "content": "从$1$、$2$、$9$、$13$中任取一个数作一个分数的分子, 从$4$、$8$、$12$、$15$中任取一个数作这个分数的分母, 可构成不同的值的真分数共有个.",
+        "objs": [],
+        "tags": [
+            "第八单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第二册计数原理例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "019129": {
+        "id": "019129",
+        "content": "若集合$A_1$、$A_2$满足$A_1 \\cup A_2=A$, 则称$(A_1, A_2)$是集合$A$的一个覆盖, 当且仅当$A_1=A_1', A_2=A_2'$时, $(A_1, A_2)$与$(A_1', A_2')$为同一个覆盖. 已知$A=\\{a, b\\}$, 那么其覆盖的总数为\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第八单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第二册计数原理例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "019130": {
+        "id": "019130",
+        "content": "从集合$A=\\{x | 1 \\leq x \\leq 20, x \\in \\mathbf{Z}\\}$中任意选取三个不同的数, 使这三个数成等差数列, 这样的等差数列共有\\blank{50}个.",
+        "objs": [],
+        "tags": [
+            "第八单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第二册计数原理例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "019131": {
+        "id": "019131",
+        "content": "如图, 一环形花坛分成$A$、$B$、$C$、$D$四块区域, 现有$4$种不同的花供选种, 要求在每块区域里种$1$种花, 且相邻 (有公共边) 的两个区域需要种不同的花, 则共有多少种不同的种植方法?\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) circle (0.2) circle (1);\n\\foreach \\i/\\j in {0/D,90/A,180/B,270/C}\n{\\draw (\\i:0.2) -- (\\i:1);\n\\draw ({\\i+45}:0.6) node {$\\j$};};\n\\end{tikzpicture}\n\\end{center}",
+        "objs": [],
+        "tags": [
+            "第八单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第二册计数原理例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "019132": {
+        "id": "019132",
+        "content": "写出从$a$、$b$、$c$、$d$四个元素中任取两个不同元素的所有排列.",
+        "objs": [],
+        "tags": [
+            "第八单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第二册计数原理例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "019133": {
+        "id": "019133",
+        "content": "将$6$本不同的书排成一排, 共有多少种不同的排法?",
+        "objs": [],
+        "tags": [
+            "第八单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第二册计数原理例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "019134": {
+        "id": "019134",
+        "content": "名学生排成两排照相, 每排$5$人, 共有多少种不同的排列方式?",
+        "objs": [],
+        "tags": [
+            "第八单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第二册计数原理例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "019135": {
+        "id": "019135",
+        "content": "从$3$、$4$、$5$、$7$、$11$、$17$这六个数字中选出$2$个数字作为一个分数的分子和分母, 共能组成多少个大小不同的分数?",
+        "objs": [],
+        "tags": [
+            "第八单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第二册计数原理例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "019136": {
+        "id": "019136",
+        "content": "一个火车站有$8$股岔道, 如果每股道只能停放一列火车, 现要停放$4$列不同的火车, 共有多少种不同的停法?",
+        "objs": [],
+        "tags": [
+            "第八单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第二册计数原理例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "019137": {
+        "id": "019137",
+        "content": "10 名学生排成两排照相, 每排$5$人, 其中甲、乙不能站在同一排, 共有多少种不同的排列方式?",
+        "objs": [],
+        "tags": [
+            "第八单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第二册计数原理例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "019138": {
+        "id": "019138",
+        "content": "甲、乙、丙、丁、戊$5$名同学参加``团员知识竞赛'', 决出第一名到第五名的名次 (无并列名次). 在所有可能的结果中, 甲是第三名且乙不是第一名的情况共有种.",
+        "objs": [],
+        "tags": [
+            "第八单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第二册计数原理例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "019139": {
+        "id": "019139",
+        "content": "用$0$、$1$、$2$、$3$、$4$、$5$这$6$个数字, 可以组成多少个没有重复数字的三位数?",
+        "objs": [],
+        "tags": [
+            "第八单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第二册计数原理例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "019140": {
+        "id": "019140",
+        "content": "7 个人站成一排. 若甲和乙不能相邻排列, 有多少种不同的排法?",
+        "objs": [],
+        "tags": [
+            "第八单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第二册计数原理例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "019141": {
+        "id": "019141",
+        "content": "要将$8$本各不相同的教科书排成一排放在书架上, 其中数学书$3$本、外语书$2$本、物理书$3$本. 如果$3$本数学书要排在一起, $2$本外语书也要排在一起, 那么有多少种不同的排法?",
+        "objs": [],
+        "tags": [
+            "第八单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第二册计数原理例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "019142": {
+        "id": "019142",
+        "content": "将$a$、$b$、$c$、$d$、$e$、$f$六个不同元素排成一列, 其中$a$不在首位, $b$不在末位. 问: 有多少种不同的排法?",
+        "objs": [],
+        "tags": [
+            "第八单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第二册计数原理例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "019143": {
+        "id": "019143",
+        "content": "用$1$、$2$、$3$、$4$、$5$这五个数字可以组成比$20000$大, 并且百位数不是$3$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}{$72$个}{$64$个}",
+        "objs": [],
+        "tags": [
+            "第八单元"
+        ],
+        "genre": "选择题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第二册计数原理例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "019144": {
+        "id": "019144",
+        "content": "某班上午有五节课, 分别安排语文、数学、英语、物理、化学各一节课, 要求语文与化学相邻, 数学与物理不相邻, 且数学课不排第一节, 则不同的排课方案共有种.",
+        "objs": [],
+        "tags": [
+            "第八单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第二册计数原理例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "019145": {
+        "id": "019145",
+        "content": "设$n$是一个不小于$17$的正整数, 用排列数表示$(n-16)(n-15) \\cdots(n-6)$.",
+        "objs": [],
+        "tags": [
+            "第八单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第二册计数原理例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "019146": {
+        "id": "019146",
+        "content": "解关于正整数$n$的方程: $\\mathrm{P}_{2 n+1}^4=140 \\mathrm{P}_n^3$.",
+        "objs": [],
+        "tags": [
+            "第八单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第二册计数原理例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "019147": {
+        "id": "019147",
+        "content": "已知$m$、$n$是正整数, 且$m \\leq n$. 求证:\\\\\n(1) $\\mathrm{P}_n^m=n \\mathrm{P}_{n-1}^{m-1}$;\\\\\n(2) $\\mathrm{P}_n^m+m \\mathrm{P}_n^{m-1}=\\mathrm{P}_{n+1}^m$.",
+        "objs": [],
+        "tags": [
+            "第八单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第二册计数原理例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "019148": {
+        "id": "019148",
+        "content": "已知$\\mathrm{P}_{10}^m=10 \\times 9 \\times \\cdots \\times 5$, 则正整数$m$的值为\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第八单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第二册计数原理例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "019149": {
+        "id": "019149",
+        "content": "下列各式中, 不等于$n$!的是\\bracket{20}.\n\\fourch{$\\mathrm{P}_n^n$}{$\\dfrac{1}{n+1} \\mathrm{P}_{n+1}^{n+1}$}{$\\mathrm{P}_{n+1}^n$}{$n \\mathrm{P}_{n-1}^{n-1}$}",
+        "objs": [],
+        "tags": [
+            "第八单元"
+        ],
+        "genre": "选择题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第二册计数原理例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "019150": {
+        "id": "019150",
+        "content": "已知$m$、$n$是正整数, 且$m \\leq n$. 求证: $\\mathrm{P}_n^n=\\mathrm{P}_n^m \\mathrm{P}_{n-m}^{n-m}$.",
+        "objs": [],
+        "tags": [
+            "第八单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第二册计数原理例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "019151": {
+        "id": "019151",
+        "content": "对于任意正整数$n$, 定义``$n$的双阶乘$n !$! ''如下:\n对于$n$是偶数时, $n ! !=n(n-2)(n-4) \\times \\cdots \\times 6 \\times 4 \\times 2$;\n对于$n$是奇数时, $n ! !=n(n-2)(n-4) \\times \\cdots \\times 5 \\times 3 \\times 1$.\n现有如下四个命题: \\textcircled{1} $(2021!!)(2022 ! !)=2022!$; \\textcircled{2} $2022 !!=2^{1011} \\cdot 1011!$; \\textcircled{3} $2022 !!$ 的个位数是$0$; \\textcircled{4} $2023 !!$的个位数是$5$. 正确的命题序号为\\blank{50}. (请写出所有正确命题的序号).",
+        "objs": [],
+        "tags": [
+            "第八单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第二册计数原理例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "019152": {
+        "id": "019152",
+        "content": "已知$n$是正整数, 求证: $\\dfrac{n}{(n+1) !}=\\dfrac{1}{n !}-\\dfrac{1}{(n+1) !}$, 并利用这一结论化简: $\\dfrac{1}{2 !}+\\dfrac{2}{3 !}+\\dfrac{3}{4 !}+\\cdots+\\dfrac{n}{(n+1) !}$.",
+        "objs": [],
+        "tags": [
+            "第八单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第二册计数原理例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "019153": {
+        "id": "019153",
+        "content": "判断下列问题分别是排列问题还是组合问题:\\\\\n(1) 从$10$名学生中任选$5$名去参观一个展览, 求有多少种不同的选法;\\\\\n(2) 从$1$、$2$、$3$、$4$、$5$这$5$个数字中, 每次任取$2$个不同的数作为一个点的坐标, 求所有不同点的个数;\\\\\n(3) 一个袋子中装有八张分别写有数字$1$、$2$、$3$、$5$、$10$、$16$、$32$、$50$的卡片. 从此袋中任取两张卡片, 问这两张卡片上的数相加所得到的不同结果共有多少种;\\\\\n(4) 有四本不同的书要分别送给四个人, 每人一本, 问一共有多少种不同的送法.",
+        "objs": [],
+        "tags": [
+            "第八单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第二册计数原理例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "019154": {
+        "id": "019154",
+        "content": "甲、乙、丙、丁$4$支篮球队举行单循环赛(即任意两支球队都要比赛一场).\\\\\n(1) 写出每场比赛的两支球队;\\\\\n(2) 写出冠亚军的所有可能情况.",
+        "objs": [],
+        "tags": [
+            "第八单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第二册计数原理例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "019155": {
+        "id": "019155",
+        "content": "判断下列问题分别是排列问题还是组合问题:\\\\\n(1) 从$6$名班委中选出$3$名作为发言代表, 共有多少种不同的选法?\\\\\n(2) 从$6$名班委中选出$3$名分别担任第一、二、三组的发言代表, 共有多少种不同的方案?",
+        "objs": [],
+        "tags": [
+            "第八单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第二册计数原理例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "019156": {
+        "id": "019156",
+        "content": "判断下列问题分别是排列问题还是组合问题:\\\\\n(1) 某校新成立舞蹈社团, 共招收了12名学生, 每两人互送一份礼物, 共送了多少份礼物?\\\\\n(2) 某校新成立舞蹈社团, 共招收了12名学生, 每两人握一次手, 共握多多少次手?",
+        "objs": [],
+        "tags": [
+            "第八单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第二册计数原理例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "019157": {
+        "id": "019157",
+        "content": "有$4$个质数: $2,3,5,11$.\\\\\n(1) 从这$4$个数中任取$2$个不同的数相加, 可以得到多少个不同的和?\\\\\n(2) 从这$4$个数中任取$2$个不同的数相减, 可以得到多少个不同的差?",
+        "objs": [],
+        "tags": [
+            "第八单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第二册计数原理例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "019158": {
+        "id": "019158",
+        "content": "从$a$、$b$、$c$、$d$这$4$个字母中任取$3$个不同的字母组成一组, 共有多少种不同的取法?",
+        "objs": [],
+        "tags": [
+            "第八单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第二册计数原理例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "019159": {
+        "id": "019159",
+        "content": "已知某个圆上有$5$个不同的点$A$、$B$、$C$、$D$、$E$, 以其中$2$个点为端点的线段共有多少条?",
+        "objs": [],
+        "tags": [
+            "第八单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第二册计数原理例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "019160": {
+        "id": "019160",
+        "content": "查阅相关资料, 了解排列、组合相关内容的发展, 以及了解历史上或者生活中有关排列、组合的故事, 谈谈自己的感悟体会, 写一篇$500$字左右的小作文. (提示: 例如可以结合``杨辉三角''的发展;``田忌赛马''的故事; 沈括在《梦溪笔谈》中关于围棋布局的讨论等. )",
+        "objs": [],
+        "tags": [
+            "第八单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第二册计数原理例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "019161": {
+        "id": "019161",
+        "content": "圆上有$10$个不同的点, 以其中任意$3$个点为顶点, 可以组成多少个不同的三角形?",
+        "objs": [],
+        "tags": [
+            "第八单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第二册计数原理例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "019162": {
+        "id": "019162",
+        "content": "某班要选举班干部, 现有$10$名候选人.\\\\\n(1) 从这$10$名候选人中任选$5$人组成班委, 共有多少种不同的选法?\\\\\n(2) 从这$10$名候选人中任选$5$人分别担任班委中五项不同的职务, 每项职务由一人担任, 每人只担任一项职务, 共有多少种不同的分配方法?",
+        "objs": [],
+        "tags": [
+            "第八单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第二册计数原理例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "019163": {
+        "id": "019163",
+        "content": "某校高中一年级举行篮球赛. 比赛时先分成两组, 其中 1 班、2 班、3 班、 4 班为第一组, 5 班、6 班、7 班、8 班、9 班、10 班为第二组. 各组先进行单循环赛(即同组中的每两支队都要比赛一场), 然后由各组的前两名共$4$支队进行单循环赛决出冠军和亚军. 问: 一共需要比赛多少场?",
+        "objs": [],
+        "tags": [
+            "第八单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第二册计数原理例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "019164": {
+        "id": "019164",
+        "content": "有甲、乙、丙三项任务, 其中甲需$2$人承担, 乙、丙各需$1$人承担. 现从$10$人中任选$4$人承担这三项任务, 不同的分配方法共有多少种?",
+        "objs": [],
+        "tags": [
+            "第八单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第二册计数原理例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "019165": {
+        "id": "019165",
+        "content": "设某地的街道把城市分割成矩形方格, 称每个方格为一个块, 小张从家里出发上班, 向东要走过$m$块, 向北要走过$n$块. 小张上班的路径中, 不同的最短路径共有多少条?\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.5]\n\\foreach \\i in {0,1,...,5}\n{\\draw (\\i,0) -- (\\i,5.5) (0,\\i) -- (5.5,\\i);};\n\\filldraw (0,0) circle (0.1) node [below left] {家};\n\\draw [->] (6,3) -- (8,3) node [right] {东};\n\\draw [->] (7,2) -- (7,4) node [above] {北};\n\\end{tikzpicture}\n\\end{center}",
+        "objs": [],
+        "tags": [
+            "第八单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第二册计数原理例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "019166": {
+        "id": "019166",
+        "content": "将$5$个不同的小球全部放入编号为$1,2,3,4$的四个盒子中, 若每个盒子中所放的球的个数不大于其编号数, 共有多少种不同的放法?",
+        "objs": [],
+        "tags": [
+            "第八单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第二册计数原理例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "019167": {
+        "id": "019167",
+        "content": "已知$m$是自然数, $n$为正整数, 且$m+1 \\leq n$. 求证: $\\mathrm{C}_n^m=\\dfrac{m+1}{n-m} \\mathrm{C}_n^{m+1}$.",
+        "objs": [],
+        "tags": [
+            "第八单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第二册计数原理例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "019168": {
+        "id": "019168",
+        "content": "计算$\\mathrm{C}_{100}^{97}$.",
+        "objs": [],
+        "tags": [
+            "第八单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第二册计数原理例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "019169": {
+        "id": "019169",
+        "content": "求满足等式$\\mathrm{C}_{n+3}^{n+1}=\\mathrm{C}_{n+1}^{n-1}+\\mathrm{C}_n^{n-2}+\\mathrm{C}_{n+1}^n$的正整数$n$.",
+        "objs": [],
+        "tags": [
+            "第八单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第二册计数原理例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "019170": {
+        "id": "019170",
+        "content": "求证: $\\mathrm{P}_m^m+\\mathrm{P}_{m+1}^m+\\mathrm{P}_{m+2}^m+\\cdots+\\mathrm{P}_{2 m}^m=\\mathrm{P}_{2 m+1}^m$, 其中$m$是正整数.",
+        "objs": [],
+        "tags": [
+            "第八单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第二册计数原理例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "019171": {
+        "id": "019171",
+        "content": "已知正整数$m$满足$\\dfrac{1}{\\mathrm{C}_5^m}-\\dfrac{1}{\\mathrm{C}_6^m}=\\dfrac{7}{10 \\mathrm{C}_7^m}$, 求$\\mathrm{C}_7^m+\\mathrm{C}_7^{m+1}+\\mathrm{C}_8^{m+2}+\\mathrm{C}_9^{m+3}+\\mathrm{C}_{10}^{m+4}$的值 (用数字作答).",
+        "objs": [],
+        "tags": [
+            "第八单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第二册计数原理例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "019172": {
+        "id": "019172",
+        "content": "已知正整数$x$、$n$满足$\\mathrm{C}_n^x=\\mathrm{C}_n^{2 x}$, 且$\\mathrm{C}_n^{x+1}=\\dfrac{11}{3} \\mathrm{C}_n^{x-1}$, 求$x$、$n$的值.",
+        "objs": [],
+        "tags": [
+            "第八单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第二册计数原理例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "019173": {
+        "id": "019173",
+        "content": "一个罐子中有大小与质地完全相同的$20$个玻璃球, 其中$4$个是红色的, $6$个是黑色的, $10$个是白色的. 经充分混合后, 从罐子中同时任取$2$个球, 求下列事件的概率:\\\\\n(1) $2$个球都是黑色的;\\\\\n(2) $2$个球的颜色不同.",
+        "objs": [],
+        "tags": [
+            "第八单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第二册计数原理例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "019174": {
+        "id": "019174",
+        "content": "在$100$件产品中有$90$件一等品、$10$件二等品, 从中随机抽取$4$件产品.\\\\\n(1) 求恰好含有$1$件二等品的概率;\\\\\n(2) 求至少含有$1$件二等品的概率;\\\\\n(3) 求至多含有$1$件二等品的概率.(以上结果均精确到$0.01$)",
+        "objs": [],
+        "tags": [
+            "第八单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第二册计数原理例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "019175": {
+        "id": "019175",
+        "content": "某学习小组共有$10$名学生, 求其中至少有$2$名学生在同一月份出生的概率. (默认每月天数相同, 结果精确到$0.001$)",
+        "objs": [],
+        "tags": [
+            "第八单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第二册计数原理例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "019176": {
+        "id": "019176",
+        "content": "甲、乙、丙、丁、战五人站成一排拍照, 求甲、乙两人不相邻的概率.",
+        "objs": [],
+        "tags": [
+            "第八单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第二册计数原理例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "019177": {
+        "id": "019177",
+        "content": "从正方体的八个顶点中随机选取三个点, 则取出的三个点为一个直角三角形的三个顶点的概率是\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第八单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第二册计数原理例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "019178": {
+        "id": "019178",
+        "content": "已知集合$J=\\{1,2,3,4,5\\}$. 现独立地随机选取集合$J$的两个非空子集$A$、$B$($A$与$B$可以相同), 求集合$A$中的最大元素小于集合$B$中的最小元素的概率.",
+        "objs": [],
+        "tags": [
+            "第八单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第二册计数原理例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "019179": {
+        "id": "019179",
+        "content": "求$(x+\\dfrac{1}{x})^5$的二项展开式.",
+        "objs": [],
+        "tags": [
+            "第八单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第二册计数原理例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "019180": {
+        "id": "019180",
+        "content": "求$(3-2 x)^6$的二项展开式中$x^3$项的系数.",
+        "objs": [],
+        "tags": [
+            "第八单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第二册计数原理例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "019181": {
+        "id": "019181",
+        "content": "利用$(a+1)^n$的二项展开式, 证明: $15^{30}-1$是$7$的倍数.",
+        "objs": [],
+        "tags": [
+            "第八单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第二册计数原理例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "019182": {
+        "id": "019182",
+        "content": "已知$(x \\sin \\theta+1)^6$的二项展开式中$x^2$项的系数与$(x-\\dfrac{15}{2} \\cos \\theta)^4$的二项展开式中$x^3$项的系数相等, 求$\\cos \\theta$的值.",
+        "objs": [],
+        "tags": [
+            "第八单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第二册计数原理例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "019183": {
+        "id": "019183",
+        "content": "用二项式定理证明: $99^{10}-1$能被$1000$整除.",
+        "objs": [],
+        "tags": [
+            "第八单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第二册计数原理例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "019184": {
+        "id": "019184",
+        "content": "已知对任意给定的实数$x$, 都有$(1+2 x)^{100}=a_0+a_1(x-1)+a_2(x-1)^2+\\cdots+a_{100}(x-1)^{100}$. 求值:\\\\\n(1) $a_0+a_1+a_2+\\cdots+a_{100}$;\\\\\n(2) $a_1+a_3+a_5+\\cdots+a_{99}$.",
+        "objs": [],
+        "tags": [
+            "第八单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第二册计数原理例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "019185": {
+        "id": "019185",
+        "content": "求$(1+3 x)^{15}$的二项展开式中系数最大的项.",
+        "objs": [],
+        "tags": [
+            "第八单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第二册计数原理例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "019186": {
+        "id": "019186",
+        "content": "若$(x^2+1)(2 x+1)^9=a_0+a_1(x+2)+a_2(x+2)^2+\\cdots+a_{11}(x+2)^{11}$, 则$a_0+a_1+a_2+\\cdots+a_{11}$的值为\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第八单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第二册计数原理例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "019187": {
+        "id": "019187",
+        "content": "在$(1+2 x)^n$的二项展开式中, 若系数最大的项是第$5$项, 求$n$的值.",
+        "objs": [],
+        "tags": [
+            "第八单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第二册计数原理例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "019188": {
+        "id": "019188",
+        "content": "辩论社团共有$11$名同学, 其中有$6$名男生, $5$名女生.\\\\\n(1) 这11名同学站成一排拍照, 在下列要求下, 分别求不同排列方法的种数:\\\\\n(I) 男生都排在一起;\\\\\n(II) 男生甲和男生乙不相邻;\\\\\n(2) 现从这$11$名同学中选出$4$名同学参加辩论比赛.\\\\\n(I) 选出的同学中男生女生都要有, 且将他们分别安排在一辩、二辩、三辩与四辩的位置上, 不同的安排方法共有多少种?\\\\\n(II) 若每位同学中选是等可能的, 则选出的同学中至少有一名是女生的概率是多少?",
+        "objs": [],
+        "tags": [
+            "第八单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第二册计数原理例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "019189": {
+        "id": "019189",
+        "content": "设$(3 x-1)^n=a_0+a_1 x+a_2 x^2+\\cdots+a_n x^n$.\\\\\n(1) 若$a_0+a_1+a_2+\\cdots+a_n=512$, 求$n$的值;\\\\\n(2) 在 (1) 的条件下, 求$x^3$项的系数$a_3$;\\\\\n(3) 若$n=12$, 求$a_1+a_3+\\cdots+a_{11}$;\\\\\n(4) 若$n=15$, 求$a_0, a_1, \\cdots, a_n$中的最大项.",
+        "objs": [],
+        "tags": [
+            "第八单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第二册计数原理例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "019190": {
+        "id": "019190",
+        "content": "某密码由$3$个数字组成, 每个数字都是$0 \\sim 9$这$10$个数字中的一个, 则该密码中, 恰有两个数字相同的概率是\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第八单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第二册计数原理例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "019191": {
+        "id": "019191",
+        "content": "现有$12$名划船运动员, 其中$2$人只会划左浆, $5$人只会划右浆, 其余$5$人双桨都会. 现要从$12$人中选取$6$人参加比赛, 要求$6$人平均分列在船舷的两侧且排定前后顺序, 问: 不同的排列方法共有多少种?",
+        "objs": [],
+        "tags": [
+            "第八单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第二册计数原理例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
     "020001": {
         "id": "020001",
         "content": "判断下列各组对象能否组成集合, 若能组成集合, 指出是有限集还是无限集.\\\\\n(1) 上海市控江中学$2022$年入学的全体高一年级新生;\\\\\n(2) 中国现有各省的名称;\\\\\n(3) 太阳、$2$、上海市;\\\\\n(4) 大于$10$且小于$15$的有理数;\\\\\n(5) 末位是$3$的自然数;\\\\\n(6) 影响力比较大的中国数学家;\\\\\n(7) 方程$x^2+x-3=0$的所有实数解;\\\\ \n(8) 函数$y=\\dfrac 1x$图像上所有的点;\\\\ \n(9) 在平面直角坐标系中, 到定点$(0, 0)$的距离等于$1$的所有点;\\\\\n(10) 不等式$3x-10<0$的所有正整数解;\\\\\n(11) 所有的平面四边形.",