Self Doubt

Question

Give a combinatorial argument to establish the identity below for any nonnegative integer n:

$\sum_{k=0}^{n} k\binom{n}{k}=n\ast 2^{n-1}$

Answer 1

> \begin{align*}  &A = \sum_{k\geq1}^{n}k\cdot \binom{n}{k} \\ \end{align*}

$A $ is the no of ways to form a committee of $k \geq 1$ people out of $n$ available individuals and select one head for the selected committee.

Empty committee is not possible. Therefore we select the head first in $n$ ways and then any subset out of $(n-1)$ remaining people in $2^{n-1} $ ways.

> \begin{align*}  &A =n\cdot2^{n-1} \\  \end{align*}