Derangement Calculator
Use this free Derangement Calculator to quickly get accurate results in seconds, free and instantly offline with no signup required ever, trusted by professionals in Austin, Chicago, Miami, Denver, Seattle, and New York for reliable daily use without accounts, subscriptions, limits, or delays on any device for everyday personal and business workflows daily.
Calculate subfactorials (derangements).
Calculator Inputs
About Derangements
Use this free Derangement Calculator to quickly get accurate results in seconds, free and instantly offline with no signup required ever, trusted by professionals in Austin, Chicago, Miami, Denver, Seattle, and New York for reliable daily use without accounts, subscriptions, limits, or delays on any device for everyday personal and business workflows daily.
How derangements work in seconds: Enter n items to get subfactorial counts where every element displaces, plus near-miss probabilities. Results appear at $0 cost in your browser.
Why derangements delight USA parties on 09/22/2026: Secret santas forbid self-draws — derangements count valid assignments exactly. Austin offices running $10,000 holiday draws generate valid loops. New York galas seating $1M donor lotteries avoid self-seating.
Exam anonymity: Chicago boards shuffling $500,000 bar-exam grading derange papers from authors. Denver labs randomizing $900,000 trial arms blind assignments properly.
Festival posts: Miami festivals assigning $200,000 volunteer shifts ensure nobody guards their own gate. Seattle QA scrambling $2M fuzz corpora avoid seed self-matches.
One-over-e limit: Large-n derangement probability converges to 36.8 percent — memorize it for party wagers.
On-call rotations: Austin startups assigning $200,000 engineer rotations derange schedules so nobody inherits their own incidents. Chicago clinics randomizing $1M patient surveys blind reviewers on 09/22/2026.
Parking lotteries: Miami condos drawing $300,000 garage assignments ensure nobody keeps their own spot unfairly. Phoenix HOAs rotating $50,000 board seats avoid entrenchment.
Gear swaps: Denver ski swaps exchanging $90,000 used-gear pools match givers to different receivers. Everyone gives, everyone gets, nobody self-deals.
Wager math: Large-n derangement probability sits at 36.8 percent — profitable trivia wagers hide in plain sight.
Exclusion proof: Inclusion-exclusion derives subfactorials cleanly — alternating sums converge fast enough for mental checks.
On-call fairness: Austin startups assigning $200,000 engineer rotations derange schedules so nobody inherits their own incidents. Chicago clinics randomizing $1M patient surveys blind reviewers on 09/22/2026.
Parking draws: Miami condos drawing $300,000 garage assignments ensure nobody keeps their own spot unfairly. Phoenix HOAs rotating $50,000 board seats avoid entrenchment systematically.
Gear swaps: Denver ski swaps exchanging $90,000 used-gear pools match givers to different receivers. Everyone gives, everyone gets, nobody self-deals ever.
Exclusion derivation: Inclusion-exclusion derives subfactorials cleanly — alternating sums converge fast enough for mental verification.
Where it is used
Austin offices run $10,000 secret-santa draws, Chicago exam boards shuffle $500,000 anonymized grading, Miami festivals assign $200,000 volunteer posts, Denver labs randomize $900,000 trial arms, Seattle QA scrambles $2M fuzz seeds, New York galas seat $1M donor lotteries, Austin startups assign $200,000 on-call rotations, Chicago clinics randomize $1M patient surveys, Miami condos draw $300,000 parking lotteries, Denver ski swaps exchange $90,000 gear pools, and Phoenix HOAs rotate $50,000 board seats.
How to use the Derangements
- Number — enter the value (e.g. 10).
- Calculate — press the Calculate button to see the result instantly above.
Formula
n! = n * (n-1) * (n-2) * ... * 2 * 1
Examples
Office draw
Card shuffle
Small group
Frequently Asked Questions
- What is a factorial and how is it calculated for derangement?
- The factorial of a non-negative integer n, written as n followed by an exclamation point, is the product of all positive integers from 1 to n. For example, 5 factorial equals 5 times 4 times 3 times 2 times 1 which equals 120. By definition, 0 factorial equals 1. The Derangement Calculator handles these calculations instantly for your derangement work.
- How quickly do factorials grow for derangement applications?
- Factorials grow faster than exponential functions. While 10 factorial is about 3.6 million, 20 factorial exceeds 2.4 quintillion. This extremely rapid growth is why factorials quickly exceed standard number representations, which is why the Derangement Calculator supports values up to 170 for derangement calculations.
- What are factorials used for in derangement applications?
- Factorials are fundamental to permutations and combinations in counting problems, probability theory, statistical distributions, Taylor series in calculus, and algorithm analysis. They appear in formulas for arrangements, lottery odds, quality control sampling, and many other practical derangement applications.
- Why is 0 factorial equal to 1 for derangement?
- The mathematical definition of 0 factorial equals 1 is necessary for consistency across many formulas. For example, the number of ways to arrange 0 items is exactly one way doing nothing. The combination formula also requires 0 factorial equals 1. This convention ensures all derangement formulas involving factorials work correctly at boundary cases.
- What is the maximum factorial value the Derangement Calculator can calculate?
- The Derangement Calculator supports factorials up to 170 factorial, which produces a number with approximately 307 digits. Values beyond 170 exceed the maximum representable number in standard floating-point arithmetic. For most practical derangement applications, even values as large as 20 factorial are rarely needed.