Question

let n be a positive integer. prove that:
1(nC1) + 2(nC2) +...+n(nCn) = n2^(n-1)

Answer 1

i'll rewrite the original expression in 2 ways (and i'll call it X):
X = 0(nC0) + 1(nC1) + ... + (n-1)(nCn-1) + n(nCn)
X = n(nCn) + (n-1)(nCn-1) + .... + 1(nC1) + 0(nC0)

now remember nCn = nC0, nC1 = nCn-1, ...
so if I add up the 2 equations I wrote, what do I get?