gen-briefly-explain-derangements-and-one-common-misconception-ab-3e2c

Briefly explain derangements and one common misconception about it.

# Derangements A **derangement** is a permutation where no element appears in its original position. For example, if you have items [1, 2, 3], the arrangement [2, 3, 1] is a derangement (1 isn't in position 1, 2 isn't in position 2, etc.), but [2, 1, 3] is not (3 is still in position 3). The number of derangements of n items is denoted D(n) and approximates n!/e for large n. ## Common Misconception Many people mistakenly think that derangements are *rare* compared to total permutations. In reality, **the probability of a random permutation being a derangement approaches 1/e ≈ 36.8%** as n grows large. This is surprisingly high! About 1 in 3 random shuffles will have nothing in its original position, which is counterintuitive since most people expect this "coincidence-free" outcome to be much more unlikely.