Hat-Check Problem

Problem
The traditional wording of the question is as follows.

A hat-check girl completely loses track of which of $$n$$ hats belong to which owners, and hands them back at random to their $$n$$ owners as the latter leave.

What is the probability $$p_n$$ that nobody receives their own hat back?

Solution
Put into bald mathematical language, this boils down to:

For a set $$S$$ of $$n$$ elements, what is the number of derangements of $$S$$ divided by the number of permutations of $$S$$?