Putnam 2007 B6

For each positive integer $n$, let $f(n)$ be the number of ways to make $n!$ cents using an unordered collection of coins, each worth $k!$ cents for some $k$, $1 \leq k \leq n$. Prove that for some constant $C$, independent of $n$, $$n^{n^2/2 - Cn} e^{-n^2/4} \leq f(n) \leq n^{n^2/2 + Cn}e^{-n^2/4}.$$