A Secret Santa Draw Fails 63 % of the Time, Whatever the Group Size

Why the answer stops depending on the group

Adding a person adds a chance that they draw themselves, and it also adds a name that everyone else might draw instead. The two effects nearly cancel, and the residue is an alternating series whose terms shrink like the reciprocals of factorials — which is exactly the series for 1/e. Because factorials grow so fast, the tail of that series is negligible after a handful of terms, and the answer freezes. Intuition expects a large group to be safer, on the grounds that one person's own name is a smaller fraction of the hat; the arithmetic says the extra opportunities for failure arrive at the same rate.

Redrawing is not the fix people think it is

Drawing again until nobody has their own name does produce a valid assignment, and on average it takes about 2.7 attempts to get there — one divided by 0.368. What it does not produce is a uniformly random valid assignment, if the redraw is done carelessly: swapping just the offending pair, or letting the person who drew themselves pick again from what is left, biases the result in ways that are invisible in a single run and obvious over a few years of the same office party. The clean approaches are to reject the whole draw and start over, or to build a single random cycle through all the names, which is a derangement by construction.

Frequently asked questions

What is a subfactorial exactly?

It counts the permutations of n objects that leave nothing fixed, written !n with the exclamation mark in front. It has a tidy recurrence — !n equals n−1 times the sum of the two previous subfactorials — and an even tidier shortcut: !n is the nearest whole number to n! divided by e, for every n from one upwards. That second fact is why the ratio in the table converges so obviously; the derangement count is essentially a factorial divided by e, rounded, and the rounding is the only thing standing between the sequence and the constant.

Does the same maths cover matching problems generally?

It covers the pure version and stops where the constraints begin. Derangements answer the question of nobody matching themselves, and nothing more: the moment a rule is added — couples must not draw each other, last year's pairs are barred, one person is away — the count changes and the neat convergence to 1/e is gone. Those variants are still countable, usually by inclusion and exclusion, but they no longer have a single memorable answer. The 36.8 per cent is a baseline for the unconstrained case, and it is worth knowing precisely because so many real draws start there before someone adds a rule.