- $\displaystyle \sum_{k=0}^n \binom{n}{k}(-1)^k = 0^n$
- $\displaystyle \sum_{k=0}^n \binom{n}{k} = 2^n$
- $\displaystyle \sum_{k=0}^n \binom{n}{k}2^k = 3^n$
- $\displaystyle \sum_{k=0}^n \binom{n}{k} k = n2^{n-1}$
- $\displaystyle \sum_{k=0}^n \binom{n}{k} k^2 =n(n+1) 2^{n-2}$
- $\displaystyle \sum_{k=0}^n \binom{n}{k}^2 = \binom{2n}{n}$
- $E_n$ を $0$ 以上 $n$ 未満の偶数全体、$O_n$ を $0$ 以上 $n$ 未満の奇数全体とする ($n\geq 1$)
このとき、$\displaystyle \sum_{k\in E_n} \binom{n}{k} = \sum_{k\in O_n} \binom{n}{k} = 2^{n-1}$
- $\displaystyle \binom{n}{k} = \binom{n-1}{k} + \binom{n-1}{k-1}$
- $\displaystyle \frac{(a+b+c)!}{a!\ b!\ c!} = \binom{a}{a}\binom{a+b}{b} \binom{a+b+c}{c}$
マイナー
- $\displaystyle \binom{n}{k} = \sum_{m=k-1}^{n-1}\binom{n}{k-1}$
- $\displaystyle \binom{n+1}{m} = \sum_{i=0}^{m} \binom{n-m+i}{i}$
- $\displaystyle\sum_{i=0}^m\frac{\binom{n}{i}}{\binom{m}{i}} = \frac{n+1}{n+1-m}$
メモ