If we have some prime $p$ and a natural number $k$, is there a formula for the largest natural number $n\_k$ such that $p^{n\_k} | k!$.

This came up while doing an unrelated homework problem, but it is not itself homework. I haven't had any good ideas yet worth putting here.

The motivation came from trying to figure out what power of a prime you can factor out of a binomial coefficient. Like $\binom{p^m}{k}$.