Secret Santa Odds Calculator
Secret Santa valid draws and odds from player count, with derangement math.
What it does
Secret Santa works only when nobody draws themselves, and the count of such match-free draws is a classic derangement. With 5 players 44 of 120 shuffles qualify, so roughly one draw in three survives.
The tool above takes the player count. It returns valid draws, the clean draw chance, and expected shuffles per success.
The default 5 players give 44 valid draws at 36.7% with 2.7 tries on average. Eight players give 14,833 at 36.8%, already at the long run rate.
Math follows Whitman combinatorics text with Applied Combinatorics inclusion-exclusion behind it. Both prove !n = round(n! / e) used here.
How to use
- Enter the player count.
- Read valid draws first.
- Read the clean draw chance beside it.
- Plan tries per the average shown.
- Redraw fully on any self match.
- Settle couples rules before drawing.
Formula
- n
- Number of players drawing.
- !n
- Valid match-free draws.
- n!
- Total permutations of n names.
- e
- Euler number 2.71828 for the limit.
Counts use the exact recurrence with !0 = 1 and !1 = 0, so large groups stay exact to 100 players. Chance settles near 1/e = 36.79% quickly. Tries is the expected number of full shuffles per clean draw.
Worked example
Example: 5 players
Defaults: 5 players drawing from a hat.
- Players: 5, so 5! = 120 permutations.
- Derangement sum: 1 - 1 + 1/2 - 1/6 + 1/24 - 1/120 = 0.3666667.
- Valid draws: 120 x 0.3666667 = 44.
- Chance: 44 / 120 = 36.7%.
- Check: round(120 / e) = round(44.15) = 44.
- Tries: 100 / 36.7 = 2.7 shuffles.
- Tool shows 44 valid draws, 36.7 %, 2.7 tries on average.
Reference table
| Players | Valid draws | Chance |
|---|---|---|
| 2 | 1 | 50.0% |
| 3 | 2 | 33.3% |
| 4 | 9 | 37.5% |
| 5 | 44 | 36.7% |
| 6 | 265 | 36.8% |
| 7 | 1,854 | 36.8% |
Limits
- Full redraws only; partial fixes differ.
- Couples exclusions shrink counts.
- Behavioral cheating sits outside math.
- Caps at 100 players.
Common mistakes
- Redrawing one slip instead of all.
- Allowing self matches quietly.
- Counting orders as distinct draws.
- Playing with one player.
- Skipping the couples rule talk.
Try it yourself
Quick checks. Answers sit tucked under each question.
8 players?
!8 = 14,833 of 40,320 = 36.8%, still 2.7 tries.
3 players?
!3 = 2 of 6 = 33.3%, 3 tries on average.
10 players?
!10 = 1,334,961 of 3,628,800 = 36.8%, 2.7 tries.
Questions
What counts as a valid draw?
A derangement: a permutation where nobody draws themselves. Count !n = round(n! / e).
What are the odds of a clean draw?
Near 36.8% past 7 players (1/e). Small groups swing: 50% at 2, 33.3% at 3.
Do we redraw on a self match?
Yes. Redraw the whole hat; the math counts full clean shuffles only.
Why not play solo?
One player cannot draw match-free, so !1 = 0 and the game needs pairs or more.
What about couples avoiding each other?
Family couples add exclusion rules that shrink valid draws below derangement counts.
How do big groups run smoothly?
Draw digitally or referee the hat so a stuck draw restarts early instead of late.