组合数有关的公式及常用求和

C m n = C m - 1 n - 1 + C m n - 1

$C_n^m = C _{n-1}^{m-1}+C _{n-1}^{m}$

m C m n = n C m - 1 n - 1

$mC_n^m = nC _{n-1}^{m-1}$

C 0 n + C 1 n + C 2 n + \dots \dots + C n n = 2 n

$C_n^0+C_n^1+C_n^2+……+C_n^n = 2^n$

1 C 1 n + 2 C 2 n + 3 C 3 n + \dots \dots + n C n n = n 2 n - 1

$1C_n^1+2C_n^2+3C_n^3+……+nC_n^n =n2^{n-1}$

12 C 1 n + 22 C 2 n + 32 C 3 n + \dots \dots + n 2 C n n = n (n + 1) 2 n - 2

$1^2C_n^1+2^2C_n^2+3^2C_n^3+……+n^2C _n^n =n(n+1)2^{n-2}$

C 1 n 1 - C 2 n 2 + C 3 n 3 + \dots \dots + (- 1) n - 1 C n n n = 1 + 1 2 + 1 3 + \dots \dots + 1 n

$\frac{C_n^1}{1}-\frac{C_n^2}{2}+\frac{C_n^3}{3}+……+(-1)^{n-1}\frac{C _n^n}{n} =1 + \frac{1}{2}+ \frac{1}{3}+……+ \frac{1}{n}$

(C 0 n) 2 + (C 1 n) 2 + (C 2 n) 2 + \dots \dots + (C n n) 2 = C n 2 n

$(C_n^0)^2+(C_n^1)^2+(C_n^2)^2+……+(C _n^n)^2 = C_{2n}^n$