TITLE: Combinatorial argument to prove the recurrence relation for number of derangements
QUESTION [18 upvotes]: Give a combinatorial argument to prove that the number of derangements satisfies the following relation:
$$d_n = (n − 1)(d_{n−1} + d_{n−2})$$
for $n \geq 2$.

I am able to prove this algebraically but not able to see the combinatorial example.

REPLY [5 votes]: This is simply the first method given in this answer. In that answer, three methods for computing the number of derangements are given.