Question 134
Let $[n]$ denote the set of integers $\{1,2, \ldots, n\}$. We randomly choose a function $f:[n] \rightarrow [n]$, out of the $n^{n}$ possible functions. We also choose an integer $a$ uniformly at random from $[n]$. Find the probability that there exist positive integers $b, c \geq 1$ such that $f^{b}(1)=a$ and $f^{c}(a)=1$. ($f^{k}(x)$ denotes the result of applying $f$ to $x$ $k$ times).