组合恒等式
组合恒等式
\[\binom n m = \binom n {n - m}
\]
\[\sum_{i = 0}^n \binom n i = 2 ^ n
\]
\[\binom n m = \binom {n - 1} {m - 1} + \binom {n - 1}{m}
\]
用最后一个调整：
\[\sum_{i = 0} ^ n \binom n i [2 | i] = 2 ^ {n - 1}
\]
k 个非负整数和为 n 的方案数（插板法） :
\[\binom {n + k - 1} { k - 1 }
\]
\[\sum_{i = 0} ^ m \binom {n + i} n = \binom {n + m + 1} m
\]
\[\sum_{i = m} ^ n \binom i m = \binom {n + 1} {m + 1}
\]
证明：下面 = 上面
\[\binom n m \binom m k = \binom n k \binom {n - k} {m - k}
\]
\[\sum_{i = 0} ^ k \binom n i \binom m {k - i} = \binom {n + m} k
\]
