MATHEMATICS · FREE TOOL

Permutation Calculator

Every permutation in one place — nPr with steps, with repetition, circular, and the letters of a word — exact answers, however huge the numbers get.

NO SIGN-UP — RUNS ENTIRELY IN YOUR BROWSER
INPUT

Permutation — P(n, r)

Arrange r of n distinct items in order, each used at most once — the classic nPr.

TRY
RESULT

ENTER VALUES — THE ANSWER APPEARS HERE LIVE

HOW TO USE

Four steps, zero friction.

No compute button, no page reloads — the calculator reworks the answer live on every keystroke.

01

Pick your mode

Five modes cover every permutation question: P(n, r) for distinct items used once, with repetition for reusable items, circular for rings, word for letters with repeats, and list to print every arrangement.

02

Enter your values

Type n and r, a word, or comma-separated symbols. Inputs accept up to n = 1000 — and the result panel updates the instant anything changes.

03

Read the working

Every answer shows the exact value, the formula with your numbers substituted, the expansion (10 × 9 × 8), and digit count plus scientific notation when numbers get huge.

04

Copy or list

Tap Copy to grab the exact answer, or switch to List them all to generate every distinct arrangement and copy the whole set in one go.

THE MATH

One idea, four shapes.

A permutation counts ordered arrangements. The only question is what your items are allowed to do — match your problem to one of these four shapes.

01 — NO REPETITION

Distinct items, used once

P(n, r) = n! / (n − r)!

Order matters and each item appears at most once — you're choosing r out of n and lining them up.

Podium from 9 runners: P(9, 3) = 9 × 8 × 7 = 504

02 — WITH REPETITION

Every slot can reuse any item

n^r

Each of the r positions independently picks from the n items — like digits in a PIN code.

4-digit PIN: 104 = 10,000 codes

03 — CIRCULAR

Around a circle, rotations equal

(n − 1)!

Fix one item to cancel rotational duplicates, then arrange the remaining n − 1 freely.

6 people at a table: 5! = 120 seatings

04 — REPEATED ITEMS

Identical items merge

n! / (a! × b! × …)

When some items are indistinguishable, divide out the internal orderings of each repeated group.

BANANA: 6! / (3! × 2!) = 60

ORDER DOESN'T MATTER? Then you need a combination, not a permutation: C(n, r) = P(n, r) / r!. The same five items give P(5, 2) = 20 ordered pairs but only C(5, 2) = 10 unordered pairs — the r! orderings of each chosen group collapse into one.
FAQ

Permutation questions, answered.

The eight questions that come up again and again when arranging things.

What is a permutation?+
A permutation is an ordered arrangement of items — a selection where the sequence matters. Choosing gold, silver and bronze from nine runners is a permutation: 9 × 8 × 7 = 504 possible podiums. Choosing any three of the nine for a photo, ignoring order, would be a combination instead.
What's the difference between a permutation and a combination?+
Order. If AB and BA count as different outcomes, it's a permutation — use P(n, r) = n!/(n−r)!. If they count as the same, it's a combination — C(n, r) = n!/(r!(n−r)!), which is always smaller because the r! orderings of each chosen group collapse into one. For example, P(5, 2) = 20 but C(5, 2) = 10.
What does nPr mean on a calculator?+
nPr is the standard notation — and the button on scientific calculators — for permutations of n items taken r at a time: nPr = n!/(n−r)!. For example, 10P3 = 10 × 9 × 8 = 720. This tool computes the same thing, plus repetition, circular and word modes the calculator button doesn't have.
How do I calculate permutations with repetition allowed?+
When each position can hold any of the n items — digits in a PIN, draws with replacement — the count is n to the power r. Ten digits across four positions gives 104 = 10,000 possible PINs. Use the “With repetition” mode above.
What is a circular permutation?+
An arrangement around a circle, where rotating the whole ring doesn't create a new outcome. Fix one item in place to cancel rotations, then arrange the remaining n − 1 items: the count is (n − 1)!. Six people around a table gives 5! = 120 seatings. If mirror images also count as the same (a necklace), it's (n − 1)!/2.
Why is 0! equal to 1?+
It's the empty product — there is exactly one way to arrange nothing. The convention keeps every formula working: P(n, n) = n!/0! = n!, and P(5, 0) = 1, since there is exactly one empty arrangement.
How large can the numbers get?+
Inputs go up to n = 1000. Results use exact big-integer arithmetic rather than floating point, so every digit is correct — 1000! runs to 2,568 digits and the calculator shows all of them, with scientific notation alongside for readability.
Does the calculator work offline and on mobile?+
Yes. It's fully responsive for phone, tablet and desktop, and all of the math runs locally in your browser — nothing is uploaded, and once the page has loaded it keeps working even if you go offline.