NULL

$ \large1.$ 容斥原理

\[f(n) = \sum_{i = 0}^n \dbinom{n}{i} g(i) \Leftrightarrow g(n) = \sum_{i = 0}^n (-1)^{n-i} \dbinom{n}{i}f(i) \]

$\large f $ 表示至多 ,\(\large g\) 表示恰好

\[f(n) = \sum_{i = n}^m \dbinom{i}{n} g(i) \Leftrightarrow g(n) = \sum_{i = n}^m (-1)^{i-n} \dbinom{i}{n}f(i) \]

​ \(\large f\) 表示至少 , $ \large g $ 表示恰好

\(\large2.\) 组合恒等式

\[\dbinom{r}{k} = \dbinom{r-1}{k} + \dbinom{r-1}{k-1} \]

\[\dbinom{r}{m}\dbinom{m}{k} = \dbinom{r}{k}\dbinom{r-k}{m-k} \]

\[\sum_{k} \dbinom{r}{k} x^k y^{r-k} = (x + y)^r \]

\[\sum_{k\leq n}\dbinom{n+k}{k} = \dbinom{r+n+1}{n} \]

\[\sum_{0\leq k \leq n} \dbinom{k}{m} = \dbinom{n+1}{m+1} \]

$ \large 3.$ 范德蒙德卷积

\[\sum_k\dbinom{l}{m+k}\dbinom{s}{n-k} = \dbinom{l+s}{m+n} \]

\[\sum_k \dbinom{l}{m+k}\dbinom{s}{n+k} = \dbinom{l+s}{l - m + n} \]

\[\sum_k \dbinom{l-k}{m}\dbinom{s+k}{n} = \dbinom{l + s + 1}{m+n+1} \]