Exercise 8.

Consider the binomial identity

\begin{equation*} \binom{n}{1} + 2 \binom{n}{2} + 3 \binom{n}{3} + \cdots + n\binom{n}{n} = n2^{n-1}\text{.} \end{equation*}
  1. Give a combinatorial proof of this identity. Hint: What if some number of a group of \(n\) people wanted to go to an escape room, and among those going, one needed to be the team captain?

  2. Give an alternate proof by multiplying out \((1+x)^n\) and taking derivatives of both sides.

Hint.
in-context