常见排列组合公式

$1:C[n][m] = f[n] / f[m] / f[n-m] (f[i]为i的阶乘)$
$2:A[n][m] = f[n] / f[m]$
$3:C[n][0]+C[n][1]+C[n][2]+...C[n][n] = 2^n$
$4:C[n][0] + C[n][2] + C[n][4] + ... = C[n][1] + C[n][3] + C[n][5] = 2^{n-1}$
$5:0*C[n][0]+1*C[n][1]+2*C[n][2]+...n*C[n][n] = n*2^{n-1}$

$证明5式：$
$(1+x)^n = C[n][0] + xC[n][1] + x^2C[n][2] + x^3C[n][3] + ... + x^nC[n][n]$
$两边求导后将x=1带入即可$