Derangements

Derangement is a type of combinatorial problem that involves permutations with no fixed points. In other words, a derangement is a permutation of a set of elements ranking from 1 to n in which none of the elements is left at its original place.

The number of derangements of a set of size n is known as the subfactorial of n or the n-th derangement number or n-th de Montmort number (after Pierre Remond de Montmort).

 

Key Concepts

  • Formula: The formula for the number of derangements D(n) of a set with n objects is

          D(n) = n![1-1/1!+1/2!-1/3!…1/N!].

Examples

  • Example 1: In how many ways can 6 people be arranged in 6 seats such that none of them are occupying their original positions?

          This is an example of a derangement problem.

          The number of ways to arrange 6 people in 6 seats is 6!.

          The number of derangements of a set of size 6 is D(6) = 265.

          Therefore, the number of ways to arrange 6 people in 6 seats

           such that none of them are occupying their original positions is

           6! × D(6) = 265,252,800

  • Example 2: In how many ways can 5 students A, B, C, D, and E grade their 5 tests such that no student grades their own test?

           This is an example of a derangement problem.

           The number of ways to grade the tests is 5!.

           The number of derangements of a set of size 5 is D(5) = 44.

           Therefore, the number of ways to grade the tests such that no student grades their own test is

           5! × D(5) = 52,800