A combinatorial sum

Identity. $\binom{n}{k}\binom{k}{2}=\binom{n}{2}\binom{n-2}{k-2}$.

Sum. $\sum_k\binom{n}{2}\binom{n-2}{k-2}=\binom{n}{2}\sum_j\binom{n-2}{j}=\binom{n}{2}\cdot 2^{n-2}=\frac{n(n-1)}{2}\cdot 2^{n-2}=n(n-1)2^{n-3}$.

Correction. The result should be $n(n-1)2^{n-3}$, not $2^{n-2}n(n-1)$.

Let me recheck: $\binom{n}{2}2^{n-2}=\frac{n(n-1)}{2}2^{n-2}=n(n-1)2^{n-3}$.

Conclusion. $\sum_k\binom{n}{k}\binom{k}{2}=n(n-1)2^{n-3}$.