How many ways are there to choose in a team of $n$ people a delegation of $k$ people, then from among them its leader and deputy? How many ways are there to choose in a team of $n$ people a delegation of $k$ people, and then from among the selected people to appoint its leader and his deputy? Try using the result to derive the formula: $$\sum_{k = 2}^{n} \binom{n}{k} = 2^{n - 2}\binom{n}{2}$$ **CORRECTION** : The formula given by the OP is incorrect. It should be $$\sum_{k = 2}^{n} \binom{n}{k}\binom{k}{2} = \binom{n}{2} 2^{n - 2}.$$

The correct formula should be $$\sum_{k = 2}^{n} \binom{n}{k}\binom{k}{2} = 2^{n - 2}\binom{n}{2}.$$ Hint. For the left side choose first the delegation and THEN appoint its leader and the deputy. For the right side choose first the leader and the deputy and THEN complete the rest of the team.