Random Fixed Point

Question

Let $f_n: [n] \rightarrow [n]$ be a uniformly random function selected from the set of all functions from $[n]$ to $[n]$, where $[n] = \{1,2,\dots,n\}$. Let $F_n$ be the number of fixed points the random $f_n$ selected. What is the limiting distribution of $F_n$? In other words, for any non-negative integer $k$, what does $\prob{F_n = k}$ converge to? Report $\displaystyle \lim_{n \rightarrow \infty} \prob{F_n = 3}$ to the nearest thousandth.