For large n, D(n) ≈ n!/e, where e ≈ 2.718.
This means about 1/e ≈ 37% of all permutations are derangements when n is large.
Example: 10 people swap hats randomly. Probability no one gets their own hat ≈ 1/e ≈ 0.368.
Derangements appear in probability and combinatorial problems.