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\contentsline {paragraph}{本文简介}{2}{paragraph*.3}
\contentsline {section}{\numberline {1}什么是组合数学}{2}{section.1}
\contentsline {paragraph}{组合数学所关心的问题}{2}{paragraph*.5}
\contentsline {paragraph}{组合数学}{2}{paragraph*.6}
\contentsline {paragraph}{数学归纳法}{2}{paragraph*.7}
\contentsline {paragraph}{组合数学解决问题的一般步骤}{2}{paragraph*.8}
\contentsline {section}{\numberline {2}排列和组合}{3}{section.2}
\contentsline {subsection}{\numberline {2.1}四个基本计数原理}{3}{subsection.2.1}
\contentsline {paragraph}{划分}{3}{paragraph*.9}
\contentsline {subsubsection}{\numberline {2.1.1}加法原理}{3}{subsubsection.2.1.1}
\contentsline {subsubsection}{\numberline {2.1.2}乘法原理}{3}{subsubsection.2.1.2}
\contentsline {paragraph}{乘法原理的第二种实用形式}{3}{paragraph*.10}
\contentsline {paragraph}{乘法原理的推广}{3}{paragraph*.11}
\contentsline {subsubsection}{\numberline {2.1.3}减法原理}{4}{subsubsection.2.1.3}
\contentsline {subsubsection}{\numberline {2.1.4}除法原理}{4}{subsubsection.2.1.4}
\contentsline {subsection}{\numberline {2.2}集合的排列}{4}{subsection.2.2}
\contentsline {subsubsection}{\numberline {2.2.1}集合的排列公式}{5}{subsubsection.2.2.1}
\contentsline {paragraph}{阶乘}{5}{paragraph*.13}
\contentsline {subsubsection}{\numberline {2.2.2}循环排列公式}{5}{subsubsection.2.2.2}
\contentsline {subsection}{\numberline {2.3}集合的组合（子集）}{5}{subsection.2.3}
\contentsline {subsubsection}{\numberline {2.3.1}集合的组合公式}{6}{subsubsection.2.3.1}
\contentsline {subsubsection}{\numberline {2.3.2}集合组合公式的推论}{6}{subsubsection.2.3.2}
\contentsline {subsubsection}{\numberline {2.3.3}组合的性质一：帕斯卡公式}{6}{subsubsection.2.3.3}
\contentsline {paragraph}{组合推理证明}{6}{paragraph*.14}
\contentsline {subsubsection}{\numberline {2.3.4}组合的性质二}{7}{subsubsection.2.3.4}
\contentsline {paragraph}{证明}{7}{paragraph*.15}
\contentsline {subsection}{\numberline {2.4}多重集合的排列}{7}{subsection.2.4}
\contentsline {subsubsection}{\numberline {2.4.1}无限重复数的多重集合排列}{7}{subsubsection.2.4.1}
\contentsline {subsubsection}{\numberline {2.4.2}有限重复数的多重集合的排列}{7}{subsubsection.2.4.2}
\contentsline {subsubsection}{\numberline {2.4.3}只有两类对象的多重集合的排列}{7}{subsubsection.2.4.3}
\contentsline {subsubsection}{\numberline {2.4.4}带标签的盒子}{8}{subsubsection.2.4.4}
\contentsline {subsubsection}{\numberline {2.4.5}非攻击性车}{8}{subsubsection.2.4.5}
\contentsline {subsection}{\numberline {2.5}多重集合的组合}{9}{subsection.2.5}
\contentsline {subsubsection}{\numberline {2.5.1}无限重复数的多重集合的组合}{9}{subsubsection.2.5.1}
\contentsline {paragraph}{证明}{9}{paragraph*.16}
\contentsline {paragraph}{上述证明过程的思考}{9}{paragraph*.17}
\contentsline {paragraph}{例子}{10}{paragraph*.18}
\contentsline {paragraph}{组合中每种类型对象出现次数的下界}{10}{paragraph*.19}
\contentsline {paragraph}{例子}{10}{paragraph*.20}
\contentsline {paragraph}{有限重复数的多重集合的组合}{10}{paragraph*.21}
\contentsline {subsection}{\numberline {2.6}有限概率}{11}{subsection.2.6}
\contentsline {paragraph}{概率}{11}{paragraph*.22}
\contentsline {section}{\numberline {3}鸽巢原理}{11}{section.3}
\contentsline {section}{\numberline {4}附录A}{11}{section.4}
\contentsline {subsection}{\numberline {4.1}数学归纳法正确性的证明}{11}{subsection.4.1}
\contentsline {paragraph}{皮亚诺公理（自然数公理）}{11}{paragraph*.23}
\contentsline {subsection}{\numberline {4.2}排列与组合}{12}{subsection.4.2}
\contentsline {subsubsection}{\numberline {4.2.1}结论“乘法原理是加法原理的推论”的证明}{12}{subsubsection.4.2.1}
\contentsline {subsubsection}{\numberline {4.2.2}补集}{12}{subsubsection.4.2.2}
\contentsline {paragraph}{相对补集}{12}{paragraph*.24}
\contentsline {paragraph}{绝对补集}{12}{paragraph*.25}
\contentsline {subsubsection}{\numberline {4.2.3}2.2.1排列公式的证明}{12}{subsubsection.4.2.3}
\contentsline {subsubsection}{\numberline {4.2.4}2.2.2循环排列公式的证明}{12}{subsubsection.4.2.4}
\contentsline {subsubsection}{\numberline {4.2.5}2.3.1 集合的组合公式的证明}{13}{subsubsection.4.2.5}
\contentsline {subsubsection}{\numberline {4.2.6}2.4.1 无限重复数的多重集合的排列公式的证明}{13}{subsubsection.4.2.6}
\contentsline {subsubsection}{\numberline {4.2.7}2.4.2 有限重复数的多重集合的排列公式证明}{13}{subsubsection.4.2.7}
\contentsline {subsection}{\numberline {4.3}其他}{13}{subsection.4.3}
\contentsline {subsubsection}{\numberline {4.3.1}算术基本定理}{13}{subsubsection.4.3.1}
\contentsline {paragraph}{算术基本定理}{14}{paragraph*.26}