组合数学公式
\(C_{n}^{m} = C_{n-1}^{m} + C_{n-1}^{m-1}\)（杨辉三角）
\(C_{r}^{r} + C_{r+1}^{r} + ... + C_{n}^{r} = C_{n+1}^{r+1}\)
\(\sum_{i=0}^{k}C_{n}^{i}*C_{m}^{k-i} = C_{n+m}^{k}\)
\(\sum_{i=0}^{n} C_{n}^{i}*x^{i} = (1+x)^{n}\)（二项式定理）
\(\sum_{i=0}^{n} (-1)^{i} * C_{n}^{i} = 0\) （上式代入 \(x=-1\)）
\(C_{n}^{0} + C_{n}^{2} + ... = C_{n}^{1} + C_{n}^{3} + ... = 2^{n-1}\)
\(C_{n+m+1}^{m} = \sum_{i=0}^{m} C_{n+i}^{i}\)
\(C_{n}^{m} * C_{m}^{r} = C_{n}^{r} * C_{n-r}^{m-r}\)
\(m * C_{n}^{m} = n * C_{n-1}^{m-1}\)
\(\sum_{i=0}^{n} C_{n}^{i}*i = n * 2^{n-1}\)
\(\sum_{i=0}^{n} C_{n}^{i}*i^{2} = n * (n + 1) * 2^{n-2}\)
\(\sum_{i=0}^{n} (C_{n}^{i})^{2} = C_{2n}^{n}\)
\(D_{n} = (n-1) * (D(n-1) + D(n-2))\)（\(D(n)\) 为错排数）
\(\sum_{k=0}^{n} C_{k}^{a} * C_{n-k}^{m-a} = C_{n+1}^{m+1}\)