1. 组合数 | 二项式
1.1. 基本性质
1.定义：$$\binom{n}{m} = \frac{n!}{k!(n-k)!}\tag{1}$$
2.递推式：$$\binom{n}{m} = \binom{n-1}{m-1} + \binom{n-1}{m} \tag{2}$$
3.奇偶性：$$\sum_{i=0}^{n} (-1)^i \binom{n}{i} = 0\tag{3}$$
4. $$m\binom{n}{m} = n\binom{n-1}{m-1}\tag{4}$$
5. $$\binom{m+n}{m} = \sum_{i=0}^{m} \binom{m}{m-i}\binom{n}{i}\tag{5.1}$$ $$\binom{2n}{n} = \sum_{i=0}^{n} \binom{n}{n-i}\tag{5.2}$$
6. $$\binom{n}{r}\binom{r}{k} = \binom{n}{k}\binom{n-r}{r-k}\tag{6}$$