I was asked to give a combinatorial proof of the following fact:
$$\forall n\in \mathbb N,\sum_{k=1}^n k\binom{n}{k} = n\times2^{n-1}$$
This seems somewhat simplistic to do via noting that the binomial theorem is: $$(1+x)^n=\sum_{k=1}^n\binom{n}{k}x^k$$ Then we can take the derivative and set x=1 to get our result above.
However, this is not via combinatorics and relies on algebra instead. How would you approach this combinatorially? I'm not sure where even to start.