# Alternative answer to choose $k$ people out of $n$, then choose 1?

The question is fairly simple, we first choose $k$ people out of $n$, that is $C(k,n)$ as the combination function, then we choose 1 person out of $k$, we have $k$ choices. The total number of choice is given as $$\sum_{k=1}^n C(k,n)\cdot k$$

I understand what it means above, however, I read a book stating that the total choice is some sort of function of $2^n$, which is made by choose 1 person out of $n$, then choose the rest $k-1$ from $n-1$, I cannot formulate anything out with $2^n$ in my results, would anybody help me on that?

Note that $k$ is not fixed, in the first formulation: you are summing over all possible sizes of group $k$. Ignoring for a moment the question of picking the special person, think about how many ways there are to pick a group from $n$ people if all sizes are allowed.