What is a derangement?

A derangement is a permutation of a set where no element appears in its original position — every single object is displaced from where it started.

What is the recurrence for !n?

!n = (n − 1) × (!(n − 1) + !(n − 2)), with base cases !0 = 1 and !1 = 0.

There's exactly one way to arrange zero objects — the empty arrangement — and it vacuously satisfies 'no object is in its original position,' so !0 = 1 by convention.

A single object has only one possible arrangement, and in that arrangement it necessarily stays in its own (only) position, so there are zero derangements of one object.

Is there a closed-form formula for !n?

Yes — !n = n! × Σ (from k=0 to n) (−1)ᵏ/k!, an alternating sum that converges quickly, which is also why !n/n! approaches 1/e as n grows large.

What is the classic 'hat-check' problem?

It asks: if n people check their hats and the hats are returned at random, in how many ways can none of them get their own hat back? The answer is exactly !n, the number of derangements of n objects.

This calculator supports n up to 1000, computed with exact BigInt arithmetic so the result stays precise no matter how many digits it has.

How is this related to factorial and probability?

As n grows, !n/n! rapidly approaches 1/e ≈ 0.3679 — meaning roughly 36.8% of all permutations of a large set are derangements, a value that stabilizes remarkably quickly even for modest n.

Derangement Calculator — Count Permutations With No Fixed Points

Our Derangement Calculator computes the subfactorial !n, the number of ways to permute n objects so that not a single one ends up in its original position, using the recurrence !n = (n − 1) × (!(n − 1) + !(n − 2)), computed with exact BigInt precision so results stay exact no matter how large n is.

Quick Answer

!n counts the permutations of n objects with no fixed points. Enter n below to instantly get the exact subfactorial via its recurrence, computed with BigInt precision.

Enter n, then click Calculate.

How to Use the Derangement (Subfactorial) Calculator — !n Online

1
Enter n, the number of objects to derange.
2
Click 'Calculate' to get the exact number of derangements !n.
3
Review the recurrence used to compute the result.

Why Use Derangement (Subfactorial) Calculator — !n Online?

A derangement is a permutation with no fixed points — every object ends up somewhere other than its original spot, the classic setup behind puzzles like 'in how many ways can n people's hats get mixed up so nobody gets their own hat back.' The count of derangements, denoted !n, satisfies a simple recurrence built from smaller derangement counts, and is closely approximated by n!/e for large n — in fact, the probability that a random permutation is a derangement converges to 1/e, a surprising appearance of Euler's number in pure combinatorics. This calculator computes !n exactly using BigInt arithmetic via the standard recurrence.

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