Prove this combinatorial identity $${n \choose k}{k \choose 1}{k-1 \choose 1} = n(n-1){n-2 \choose k-2}$$

$${n \choose k}{k \choose 1}{k-1 \choose 1} = \frac{n!}{(n-k)!k!}\cdot k\cdot (k-1)=\frac{n!}{(n-k)!(k-2)!}\\=n\cdot(n-1)\cdot \frac{(n-2)!}{(n-2-(k-2))!(k-2)!}=n(n-1){n-2 \choose k-2}$$
