EXAMPLE 40 Counting permutations: Secret Santas

“Secret Santa” refers to a method whereby each member of a group anonymously buys a holiday gift for another member of the group. Each person is secretly assigned to buy a gift for another randomly chosen person in the group. Suppose Jessica, Laverne, Samantha, and Luisa share a dorm suite and want to do Secret Santa this holiday season.

  1. Verify that in this instance one woman purchasing a gift for another woman represents a permutation.
  2. Calculate how many possible different permutations of gift buying there are for the four women.

Solution

    • There are women, and people are associated with each gift (the giver and the receiver).
    • Each person can buy only one gift, so repetition is not allowed.
    • Finally, there is a difference between Jessica buying for Laverne and Laverne buying for Jessica. Thus, order is important, and thus, buying a gift represents a permutation.

  2. The number of permutations is calculated as follows: