Combinatorial proof of this question?

Question

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.

Answer 1

How many ways are there from a group of $n$ people to fix a committee of arbitrary size and elect its head?

Overall count - pick the head in $n$ ways, and the rest $n-1$ can participate or be excluded, 2 choices for $n-1$ people, a grand total of $n2^{n-1}$ ways.

Alternatively, for a committee of size $k$, pick $k$ people in $\binom{n}{k}$ ways and fix the head in $k$ ways...