Exercise C.1.15

$\star$ Show that for any integer $n \ge 0$,

$$ \sum_{k=0}^n\binom{n}{k}k = n2^{n-1} $$

$$ \sum_{k=0}^n\binom{n}{k}k = \sum_{k=1}^n\binom{n}{k}k = \sum_{k=1}^n\frac{nk}{k}\binom{n-1}{k-1} = n\sum_{k=1}^n\binom{n-1}{k-1} = n\sum_{k=0}^n\binom{n-1}{k} = n2^{n-1} $$