Letters in wrong envelopes

What this derangement probability question tests

This is a medium-difficulty probability question that asks you to compute the probability that a random permutation has no fixed points—a concept known as a derangement. Quant firms use this to assess whether you can move fluidly between combinatorial counting and probabilistic reasoning.

To solve it, you need to count the number of ways to arrange n objects such that none occupies its original position, then divide by the total number of arrangements. The problem also asks you to recognise the limiting behaviour as n grows large, which connects discrete combinatorics to continuous probability.

• Inclusion-exclusion principle for counting derangements
• Permutations and fixed points
• Asymptotic behaviour and convergence to 1/e