Prove using binomial expansion that $\sum _{k=0}^n\:k\:\begin{pmatrix}n\\ k\end{pmatrix}=n\:2\:^{n-1},\:\forall \:n\in \mathbb{Z},\:n\ge 1$

Question

I have no idea how to prove that

$\sum _{k=0}^n\:k\:\begin{pmatrix}n\\ k\end{pmatrix}=n\:2\:^{n-1},\:\forall \:n\in \mathbb{Z},\:n\ge 1$

What I would do is to prove via induction but they ask me to use binomial expansion, I don't know how to start proving.

Answer 1

It's an old trick: Do the binomial expansion of $f(x)=(1+x)^n$, then you derive it. You then set $x=1$ and you are done !