Hat Matcher V

$n$ men attend a party. Since it is formal, there is a hat closet at the entrance. All $n$ men put their distinct hats into the closet at the beginning of the party. At the end, the power at the party goes out! Each man goes into the closet and selects a hat uniformly at random. Let $q(k,n)$ be the the probability that exactly $k$ of the $n$ men receive their hat back. For a fixed $k \geq 0$, compute $\displaystyle \lim_{n \rightarrow \infty} q(k,n)$. Evaluate this for $k = 3$ to the nearest thousandth.