Common permutation and combination formulas

$1: C [n] [m] = f [n] / f [m] / f [nm] (f [i] is the factorial i)$
$2:A[n][m] = f[n] / f[m]$
$3:C[n][0]+C[n][1]+C[n][2]+...C[n][n] = 2^n$
$4:C[n][0] + C[n][2] + C[n][4] + ... = C[n][1] + C[n][3] + C[n][5] = 2^{n-1}$
$5:0*C[n][0]+1*C[n][1]+2*C[n][2]+...n*C[n][n] = n*2^{n-1}$

$5 proven formula:$
$(1 + x) ^ n = C [n] [0] + xC [n] [1] + x ^ 2C [n] [2] + x ^ 3C [n] [3] + ... + x ^ nC [n] [n]$
$After Derivative sides can be brought into x = 1$