Find sum of binominal formula and prove it

Question

I have hard time with finding sum of this:

$$ \sum_{k=1}^{n}k{n\choose k} $$

Please help! Prferably with some good hints.

Answer 1

Hint. Let define : $$f:x\in\mathbb{R}\mapsto (1+x)^n.$$ Compute $f'(1)$ using two different ways.

Answer 2

Hint. Recall that $\binom nk$ counts the $k$-subsets of an $n$-set. So $$ \sum_{k=1}^n k \binom nk = \left|\left\{(A,x) : x \in A, A \subseteq \{0,\ldots, n-1\right\}\right| $$ Counting in another way (first choosing $x$, then the set $A$), we have $$ \sum_{k=1}^n k\binom nk = n \sum_{k=1}^{n} \binom{n-1}{k-1} $$ (and the last sum can be computed easily).