Secret Santa Pairing Calculator

Work out secret santa pairing instantly with clear inputs, formula shown and shareable results.

About Secret Santa Pairing Calculator

Valid Secret Santa assignments are derangements, permutations with no fixed point, counted by the recurrence D(n) = (n - 1) x (D(n-1) + D(n-2)). The share of valid draws converges on 1 over e, about 36.8%, so a naive draw from a hat fails roughly two thirds of the time regardless of group size.

Formulas

Derangements

D(n) = (n - 1) x (D(n-1) + D(n-2)), with D(1) = 0 and D(2) = 1

Valid share

share = D(n) / n!, which tends to 1/e

Worked example

With the default values below, Secret Santa Pairing Calculator returns Valid pairings with nobody drawing themselves of 479,001,600. Change any input above and every figure here is recalculated in your browser.

Inputs used

Number of participantsaccepts 2–20: 12
Gift budget eachminimum 0.01: 20

Results

Valid pairings with nobody drawing themselves: 479,001,600
Total possible pairings: 479,001,600
Share of draws that are valid: 100.0000%
Total spend across the group: $240.00

Step by step

1Values used
2Derangements
3Valid share
4Valid pairings with nobody drawing themselves= 479,001,600
5Total possible pairings= 479,001,600
6Share of draws that are valid= 100.0000
7Total spend across the group= 240.00

Assumptions and limits

The result is derived only from the 2 inputs listed above. Anything not among them is not part of the calculation.
In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
Figures are rounded for display to 4 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

How is the Secret Santa Pairing figure worked out?

D(n) = (n - 1) x (D(n-1) + D(n-2)), with D(1) = 0 and D(2) = 1. Valid Secret Santa assignments are derangements, permutations with no fixed point, counted by the recurrence D(n) = (n - 1) x (D(n-1) + D(n-2)).

Why does this matter?

The share of valid draws converges on 1 over e, about 36.8%, so a naive draw from a hat fails roughly two thirds of the time regardless of group size.

What do I need to enter?

Enter number of participants, gift budget each. The defaults shown are a realistic worked example — swap in your own figures to get a result you can use.