The Mind-Boggling Statistics of Secret Santa: 20 Individuals, 2.4 Quintillion Potential Outcomes

The office Secret Santa or Kris Kringle can be an awkward social event for many people. It involves employees randomly buying gifts for their colleagues, often resulting in wildly inappropriate presents. The sheer statistical improbability of these gifts being passed around the office is like a Christmas miracle.

To calculate the number of possible pairings of buyers and recipients, you need to consider the permutations of the people involved. For example, in a workplace with four employees, there are 24 possible permutations (4!). As the number of employees increases, the number of permutations becomes mind-bogglingly large.

However, what Secret Santas really want is a derangement, where no one has to buy their own gift. The number of ways employees can be assigned another unique co-worker is called the de Montmort number, which is equal to n!/e (rounded to the nearest whole number). This reduces the number of permutations significantly.

Another interesting feature of Secret Santa is the number of people who will, on average, be assigned their own name in a random draw. Regardless of the number of participants, the expected number of self-matches remains the same – just one person.

If you find yourself participating in a workplace Secret Santa, hope that you receive something useful and not an inappropriate gift. But ideally, it would be best to avoid being dragged into a Secret Santa altogether.