Let's begin by writing:

\(\displaystyle S=\sum_{k=2}^{n}\left(k(k-1){n \choose k}\right)=\sum_{k=2}^{n}\left(k(k-1)\frac{n!}{k!(n-k)!}\right)=\sum_{k=2}^{n}\left(\frac{n!}{(k-2)!(n-k)!}\right)\)

Re-index:

\(\displaystyle S=\sum_{k=0}^{n-2}\left(\frac{n!}{k!(n-(k+2))!}\right)=\sum_{k=0}^{n-2}\left(\frac{n(n-1)(n-2)!}{k!((n-2)-k)!}\right)\)

Can you proceed?