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