Counting Derangements in Combinatorics

Derangements are permutations with no fixed points, meaning no element gets mapped to itself. We use the principle of inclusion-exclusion to derive as simple as possible a formula for the number of derangements of a given finite set. Amazingly, it turns out that, as the number of elements goes to infinity, the fraction of permutations that are derangements approaches the reciprocal of Euler’s constant $e$. Wow!