Induction

For any positive integer n and any non-negative integer k =< n, denote the coefficient of x^k in the expansion of (1+x)^n by nCk.
(a) It is given that nCk + nC(k+1) = (n+1)C(k+1).
Prove by mathematical induction that

kCk + (k+1)Ck + (k+2)Ck + ... + nCk = (n+1)C(k+1)
By considering the derivative and the expansion of [(1+x)^n - 1]/x , prove that

nC2 + 2 nC3 + 3 nC4 + ... + (n-1) nCn = (n-2)2^(n-1) + 1.