知乎的一个竞赛生写的排列组合介绍https://www.zhihu.com/question/26094736
二项式
注意:Cn0=1,什么都不选,是一种选择。按照定义Cn0=n!0!∗(n−0)!注意:C_{n}^0=1,什么都不选,是一种选择。按照定义C_{n}^0=\frac{n!}{0!*(n-0)!}注意:Cn0=1,什么都不选,是一种选择。按照定义Cn0=0!∗(n−0)!n!
an表示第n个括号的a,bn表示第n个括号的ba_{n}表示第n个括号的a,b_{n}表示第n个括号的ban表示第n个括号的a,bn表示第n个括号的b
a1+b1=a1+b1(1)a_{1}+b_{1}=a_{1}+b_{1}\quad\quad (1)a1+b1=a1+b1(1)
(a1+b1)(a2+b2)=(a1+b1)a2+(a1+b1)b2(a_{1}+b_{1})(a_{2}+b_{2})=(a_{1}+b_{1})a_{2}+(a_{1}+b_{1})b_{2}(a1+b1)(a2+b2)=(a1+b1)a2+(a1+b1)b2
=a1a2+b1a2+a1b2+b1b2(2)\quad\quad\quad\quad=a_{1}a_{2}+b_{1}a_{2}+a_{1}b_{2}+b_{1}b_{2}\quad\quad(2)=a1a2+b1a2+a1b2+b1b2(2)
(a1+b1)(a2+b2)(a3+b3)=(a1a2+b1a2+a1b2+b1b2)(a3+b3)(a_{1}+b_{1})(a_{2}+b_{2})(a_{3}+b_{3})=(a_{1}a_{2}+b_{1}a_{2}+a_{1}b_{2}+b_{1}b_{2})(a_{3}+b_{3})(a1+b1)(a2+b2)(a3+b3)=(a1a2+b1a2+a1b2+b1b2)(a3+b3)
=(a1a2+b1a2+a1b2+b1b2)a3+(a1a2+b1a2+a1b2+b1b2)b3=(a_{1}a_{2}+b_{1}a_{2}+a_{1}b_{2}+b_{1}b_{2})a_{3}+(a_{1}a_{2}+b_{1}a_{2}+a_{1}b_{2}+b_{1}b_{2})b_{3}=(a1a2+