Derangement (Subfactorial) Calculator
Calculate !n, the number of derangements of n elements — permutations that leave no item in its original position. Returns the exact count, n!, and the derangement probability, computed with arbitrary-precision arithmetic.
Input
Output
| Metric | Value |
|---|---|
| No data yet | |
More ways to use this tool
REST API
curl -X POST https://api.iotools.cloud/v1/tool/derangement-calculator \
-H "Authorization: Bearer YOUR_API_KEY" \
-H "Content-Type: application/json" \
-d '{
"n": "5"
}'Swap in your own key from your account. The tool's fields are the body — no wrapper.
Ask an AI agent
Use the IOTools `derangement-calculator` tool (Derangement (Subfactorial) Calculator) on this input:
YOUR_INPUT_HEREPaste this at any agent connected to the IOTools MCP server, then add your input.
Embed widget
<iframe
src="https://iotools.cloud/embed/derangement-calculator/"
width="100%" height="520" frameborder="0" scrolling="no" loading="lazy"
title="Derangement (Subfactorial) Calculator — iotools.cloud"
sandbox="allow-scripts allow-forms allow-same-origin allow-downloads allow-popups allow-popups-to-escape-sandbox"
allow="clipboard-write"
style="width:100%;border:1px solid #e5e7eb;border-radius:12px;overflow:hidden"></iframe>
<script src="https://iotools.cloud/embed.js" async></script>Drop this into your own page — free, no key required, just a link back.
| Cost per API/MCP call | From 5 credits |
|---|---|
| Need more credits? | View pricing |
Also available with
Guides
The Derangement (Subfactorial) Calculator computes !n, the number of derangements of n elements — permutations of a set that leave every element out of its original position. Enter n and it instantly returns the exact count, the total number of permutations (n!), and the derangement probability (!n / n!), computed with arbitrary-precision arithmetic so results stay exact even for large n.
It's built for combinatorics students, teachers preparing worked examples on the classic "hat-check problem," and anyone who needs an exact derangement count without hand-cranking factorials.
How to use it
- Enter n, the number of elements (0 or greater).
- Read the Derangements !n row for the count, Total permutations n! for the full factorial, and Probability (!n / n!) for the fraction of permutations with no fixed points.
- Step-by-Step Solution shows the recurrence used to build up the answer.
Results update automatically as you type. For example, entering n = 5 returns !5 = 44 out of 5! = 120 total permutations — a probability of about 0.3667.
What is a derangement?
A derangement is a permutation where no element ends up in its own original position. The name comes from the classic "hat-check problem": if n people check their hats and the hats are returned at random, a derangement is an outcome where nobody gets their own hat back.
How is !n calculated?
This calculator uses the standard recurrence relation, built up iteratively from the base cases:
!0 = 1
!1 = 0
!n = (n − 1) × (!(n − 1) + !(n − 2)) for n ≥ 2This is equivalent to the closed-form sum !n = n! × Σ(k=0 to n) (−1)^k / k!, but the recurrence avoids floating-point rounding entirely — every intermediate value is an exact integer.
Why does the probability approach 1/e?
As n grows, the derangement probability !n/n! converges to 1/e ≈ 0.367879 — a surprisingly fast convergence; by n = 10 the probability already matches 1/e to six decimal places. This means that no matter how large the hat-check line gets, there's roughly a 37% chance nobody gets their own hat back.
Need permutations or combinations instead — arrangements that don't have the no-fixed-point restriction? See the Permutation & Combination Calculator, which computes nPr and nCr from the same n! building block.
Privacy
This calculator runs entirely in your browser. Your input is never uploaded, logged, or stored — the computation happens locally on your device.