A derangement is a permutation where no element appears in its original position. D(n) = n! * [1 - 1/1! + 1/2! - 1/3! + ... + (-1)^n/n!]. Quick values: D(1)=0, D(2)=1, D(3)=2, D(4)=9, D(5)=44. Also satisfies recurrence: D(n) = (n-1)[D(n-1) + D(n-2)]. JEE application: letters in wrong envelopes, hats to wrong people.