7.2.2 计数中的常见转化

计数中的常见转化

一些组合恒等式

\[\binom n m = \binom {n} {n - m} \\ \sum_{i = 0}^n \binom n i = 2^n \\ \sum_{i = 0}^n \binom i m = \binom {n + 1} {m + 1} \\ \sum_i \binom n i \binom m {k - i} = \binom {n + m} k \\ m\binom n m = n \binom {n - 1} {m - 1} \]

最后一个式子推广一下，可以转化成“吸收公式”

\[m^{\underline{k}} \binom n m = n^{\underline{k}} \binom {n - k} {m - k} \]

普通幂和下降幂的转化

\[x^n = \sum_{k = 0}^n {n \brace k} x^{\underline k} \]