discrete math question

Question

prove that the sum of k C(n,k), 1<=k<=n is n x 2^(n-1)

Answer 1

Using the binomial expansion

(1+x)ⁿ = C(n,0) +C(n,1)x + ... + C(n,k)x^k + ... + C(n,n)xⁿ, and taking derivatives on both sides would yield

n(1 + x)^n-1 = C(n,1) + 2C(n,2)x + ... + kC(n,k)x^k-1 + nC(n,n)x^n-1.

Letting x =1 does the job.