Expected letters reaching the right firm at random

You have written $N$ cover letters, one for a recruiter at each of $N$ firms, but have lost track of which letter belongs to which firm. You send exactly one letter to each firm, matching them by a uniformly random permutation. What is the expected number of letters that arrive at the firm they were written for?

$1$. Each letter lands at its intended firm with probability $\tfrac1N$. Summing these indicator expectations over all $N$ letters, which needs no independence, gives $N\cdot\tfrac1N=1$ for every $N$.