Derangements

D(n): permutations where no element appears in its original position

D(n) = n! Σ_{k=0}^{n} (−1)^k / k! ≈ n!/e | D(n) = (n−1)(D(n−1)+D(n−2))

The hat-check problem: n guests leave hats, attendant returns them randomly. As n→∞, the probability all are wrong → 1/e ≈ 36.79%. Remarkably fast convergence — already within 0.1% by n=5.