排列组合（9月8日更新

文章目录

1.排列组合两个原理2.排列2.1.有放回2.2.无放回全排列（特殊情况） 3.组合3.1.二元3.2.多元3.3.分块取3.4.有重复组合数二项式公式

1.排列组合两个原理

$A_1$事件$n_1$种方法，$A_2$事件$n_2$种方法。

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2.排列

2.1.有放回

n个元素有放回得取r个元素排列为有重复的排列–>种类有$n^r$种

就相当于在大小为n得序列中随机选一个元素选择r次

2.2.无放回

n个元素无放回得取r个元素排列为有重复的排列–>种类有$A_n^r$种 $A_n^r=n(n-1)(n-2)...(n-r+1)=\frac{n!}{(n-r)!}$ 特指：n==r,此类排列叫全排列

无放回取r次，第一次有n种选择，接下来有n-1次，以此类推，到第r此选择得时候还有n-r+1个元素，所以就有n-r+1个可能性

全排列（特殊情况）

$P_n=n(n-1)\times (n-2)\times ...\times 3\times 2\times 1=n!$

3.组合

3.1.二元

n个不同元素抽出r个不考虑顺序成为组合，总数为： $C_n^r=\frac{A_n^r}{r!}=\frac{n(n-1)(n-2)...(n-r+1)}{r!}=\frac{n!}{(n-r)!r!}$

其实这就是常见的二项式系数，例如$(a+b{)}^n={\sum }_{r=0}^n{C}_n^r{a}^r{b}^{n-r}$

3.2.多元

3.1下一步得拓展： $r_1+r_2+...+r_k=n$,把n个元素分成k各部使得$n_i=r_i(i\in [1,k])$,结果为$\frac{n!}{r_1!r_2!...r_k!}$

计算方法： 1.分成二元时结果为$C_n^r$，将二元元素看作$r_1$和$r_{teage{t}_1}(r_1+r_{teage{t}_1}=n)$,将组合公式看成$C_n^{r1}$或$C_n^{r_{teage{t}_1}}$ 2.第二部分$r_{teage{t}_1}$继续分成二元$r_2$和$r_{teage{t}_2}(r_2+r_{teage{t}_2}=r_{teage{t}_1})$成为$C_{teage{t}_1}^{r_2}$或$C_{teage{t}_1}^{teage{t}_2}$，这时可以看成把n个元素分成三部分$C_n^{r1}{C}_{teage{t}_1}^{r_2}=C_n^{r_1}{C}_{n-r_1}^{r_2}$，化简后得到：$\frac{n!}{(n-r_1)!r_1!}\times \frac{(n-r_1)!}{(n-r_1-r_2)!r_2!}=\frac{n!}{r_1!r_2!r_{teage{t}_2}!}$，把$r_{teage{t}_2}$看成$r_3$可将最终三元答案记为$\frac{n!}{r_1!r_2!r_3!}$ 3.不严格的数学归纳一下，答案就出来了…(这里公式难打所以我们先这样做，规范化证明以后放出)

3.3.分块取

$r_1+r_2+...+r_k=r$,把n个元素分成k各部使得${\sum }_{i=1}^k{n}_i=n$,需要取得r个元素，且要从$n_i$中取得$r_i$个，

简化思想，分开拿 直接每块拿一个照写 答案为：${\prod }_{i=1}^k{C}_{n_i}^{r_i}$

3.4.有重复组合数

n个元素有重复的取r个，答案为:$C_{n+r-1}^r$

$C_n^1$看做有重复的取一个 $C_{n+1}^2$就可以类推成有重复的拿两个，因为重复导致可以看作将n个中随机拿出一个复制一次然后放进去一次抽去两个 以此类推 最后答案就是上面的答案了

二项式公式

$0!=1$ $C_n^k=C_n^{n-k}$ $(1+x{)}^n={\sum }_{r=0}^n{C}_n^r{x}^r$，上式x=1时，${\sum }_{r=0}^n{C}_n^r=2^n$ $C_a^0{C}_b^n+C_a^1{C}_b^{n-1}+..+C_a^n{C}_b^0=C_{a+b}^n$ 证明挖坑 改日更新