Permutations vs combinations — when do I use which?

Permutations care about order (gold/silver/bronze on a podium = 3 different positions). Combinations don't (picking a 3-person committee from 10 — order doesn't matter). For the same n and r, P(n,r) is always larger than C(n,r) by a factor of r!.

By convention and definition. It's the empty product (just like x^0 = 1 — the product of no x's). Makes the recursive definition n! = n × (n-1)! work for n = 1 too. Also makes combinations formulas degenerate cleanly when r = 0 or r = n.

Why does 170! overflow?

Native JavaScript Number can represent integers exactly up to 2^53. 170! ≈ 7.3 × 10^306 — still below MAX_VALUE (10^308) but with precision loss. The tool uses BigInt for exact integer math up to any size; larger n falls back to floating-point approximation.

factorial · permutation · combination · bigint · local

Compute n!, P(n,r), C(n,r)

Exact arbitrary-precision results via BigInt for n ≤ 170, Stirling approximation above. Permutations count ordered arrangements; combinations count unordered subsets.

Factorial · Permutations · Combinations

Function

kind

size

—

n only

Digits

exact

input · n and r

Exact BigInt for n ≤ 170. Above that, Stirling's approximation gives mantissa × 10^exponent.

output · value

10!

3628800

7 digits

Quick Start

Choose the right function

n! counts orderings of n items. P(n,r) counts ordered selections of r from n. C(n,r) counts unordered selections.

01
Pick function
Factorial, permutations, or combinations.
Inputs are treated as UTF-8 throughout the pipeline and validated client-side as you type — nothing dispatches until the value is well-formed.
opinpututf-8client-sidevalidated
op · textC(n,r) · set
```
combinations
```
loaded · parsedclient-side
02
Set n
Total items. Up to 170 for exact; above is approximate.
Each toggle re-computes through a memoised hook, so flipping options never resets your input and the UI stays snappy on large inputs.
noptionslivememoisedtoggles
n · memoised52 · live
```
deck of cards
```
options set · memoisedlive
03
Set r
How many to pick (skipped for factorial).
Randomness — where used — comes from crypto.getRandomValues, never Math.random. Heavy work sits in useMemo.
rcomputeinstantlocaldeterministic
r · instant5 · computed
```
poker hand
```
computed · instantinstant
04
Read result
Exact BigInt or Stirling estimate.
Copy writes through navigator.clipboard and downloads use a Blob — zero network traffic. Same input plus same options always yield the same bytes.
readcopydownloadzero-networkreproducible
ok · copiedready · copied
```
2,598,960
```
shipped · ready to shipsuccess

Pro tips

Pro tips for everyday utilities

Reach for crypto.getRandomValues

Cryptographically random for IDs, tokens, salts. Math.random is for animation and games, not security.

Sort IDs lexicographically

UUID v4 has no time ordering. Use ULID or UUID v7 if you need sorted IDs.

Mock data with seeds

Faker-style libraries can be seeded so tests are reproducible. Embed the seed in the fixture file.

Reference cards beat memory

Cheatsheets for Vim, Git, Docker, Linux signals are faster than man pages when you’re mid-flow.

Browser APIs vary

Touch events, gamepads, and clipboard differ across vendors. Feature-detect before use.

Persist in the right place

localStorage for user preferences, sessionStorage for transient state, IndexedDB for structured data.

Use Cases

When to count combinatorially

Probability, lottery odds, scheduling.

Poker probabilities

Count poker hands via C(52,5). Random values come from crypto.getRandomValues — cryptographically secure for tokens, salts, and session IDs. Math.random is deterministic enough for animations but predictable enough to be a security bug.

CSPRNG-backed

Lottery odds

C(49,6) for 6/49 — the denominator of your win chance. Pair with hashids, ULID, or UUID v7 when you need both random and sortable — useful for primary keys, public-facing IDs, and event log ordering without a sequence column.

reproducible

Race ordering

P(8,3) for podium permutations in 8-racer field. Output is deterministic given the same seed, so fixtures and tests stay reproducible across CI runs and contributor machines. Embed the seed in the fixture file for full replay.

snappy

Test combinations

C(k, n) for QA test selection. Heavy work runs through useMemo so the UI stays snappy on large inputs. Same input + same options yields identical bytes — the kind of stability snapshot tests rely on.

feature-detected

Scheduling

n! arrangements of n meetings. Storage tools persist to the right backend: localStorage for user preferences, sessionStorage for transient state, IndexedDB for structured data. Pick the right shelf and quota worries disappear.

deterministic

Birthday paradox

1 - 365! / (365−n)! / 365^n for n people. Browser feature detection: clipboard, fullscreen, gamepad, and Web Bluetooth APIs gracefully fall back when unavailable. No surprise white screens on Safari or older Firefox.

production-ready

FAQ

Frequently asked questions

Why 170 cap for exact?

170! is the largest factorial fits in a double; BigInt is exact, but above 170 the digits compound dramatically. We still compute symbolically up to ~5000, but display switches to scientific past that for readability.

Permutation vs combination?

Order matters in permutations (P), not in combinations (C). P(n,r) = n!/(n−r)!; C(n,r) = n!/(r!(n−r)!).

Is r=0 allowed?

Yes. P(n,0) = 1 and C(n,0) = 1.

What's Stirling?

An asymptotic expansion of ln(n!) used when exact factorial is impractical.

Does it run locally?

Yes — all math runs in your browser with BigInt. Nothing is sent to a server.

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