I need help to simplify this sum:
$$ \sum_{k=1}^{n} \binom{n}{k} ((k-1)!)^2 \left[ \binom{n}{k} - \binom{n-k}{k} \right]$$
I have tried to use Pascal's formula for the difference: $$\binom{n}{k} - \binom{n-k}{k} = \sum_{p=1}^{k} \binom{n-p}{k-1} $$
But i didn't see anything to simplify. Is it possible to simplify it ?
Thank you.