Permutation Calculator Online — Free Permutation Calculator Tool
CodexFuse Team · 2026-11-03T15:45:44.412Z · 8 min read min read
Use the free permutation calculator to work out ordered arrangements in seconds — nPr, circular permutations and derangements, with a worked example.
A permutation calculator answers one precise question: how many ordered arrangements can you make when you choose r items from n distinct items and the order matters? If first, second and third places are different outcomes, you are counting permutations — and the count grows far faster than most people expect. The free CodexFuse tool works it out instantly, so you can stop multiplying factorials by hand.
Permutation questions hide inside rotas, seating plans, password policies and competition draws — usually when someone needs a reliable number before a meeting starts. Guessing, or quietly using the wrong formula, produces a plan built on a number that was never right.
What a permutation actually counts
A permutation is an ordered arrangement. Take three people — Amira, Bilal and Chen — filling three distinct roles. Amira first, Bilal second, Chen third is one outcome. Bilal first, Amira second, Chen third is another, even though the same three people are involved. The formula is P(n,r) = n! / (n − r)!, which is simply the top r factors of n! multiplied together.
- Order changes the answer. Swapping two items creates a new arrangement, so permutations always outnumber combinations of the same items.
- Each item is used once, unless you say otherwise. Standard nPr never reuses an item in the same arrangement.
- The count explodes quickly. P(10,3) is 720, but P(52,5) — ordered five-card hands — is 311,875,200.
Before you enter any numbers, say the outcome out loud in one sentence: "I am arranging r distinct items, in order, from n items." If that sentence is wrong, the formula is wrong too, no matter how accurate the arithmetic is.
When does order matter — and when does it not?
Order matters whenever swapping two items creates a different outcome. First, second and third places in a race are different results even with the same three runners, so you count permutations. If only the group matters — the three people on a committee, regardless of who is listed first — you count combinations instead, and the answer is always smaller.
The test is simple: swap two items and ask whether anything real has changed. If the winner, schedule or code changes, order matters. If not, it does not.
| Situation | Order matters? | Count this |
|---|---|---|
| Podium places in a race | Yes — places are distinct | Permutation |
| Three people on a committee | No — only membership counts | Combination |
| A 4-digit PIN, digits reusable | Yes — digit position counts | Permutation with repetition |
| Seating around a round table | Yes — but rotations repeat | Circular permutation |
How to use the free permutation calculator
The workflow is short, and every field does exactly what its label says. These steps were verified against the live tool page:
- Choose a type. Pick nPr for ordinary arrangements, Circular for round-table seating, or Derangement for arrangements where nothing stays in place.
- Enter n and r. Type the total items (n) and how many you arrange (r) — or tap a preset: P(52,5), P(10,3), P(5,5) or P(8,3).
- Set repetition. Switch on "Repetition allowed" only when items can be reused; the count then becomes n to the power of r.
- Copy the result. The answer appears with thousands separators, ready to copy with one tap.
Inputs are whole numbers with n between 0 and 1000, and r cannot exceed n — the tool says so plainly if you try. P(n,0) equals 1 for any n: exactly one way to arrange nothing.
A worked example: three distinct roles from ten volunteers
Fictional scenario. Sara runs a school science fair, has ten volunteers, and opens the permutation calculator. She must fill three distinct roles — lead presenter, demo assistant and timekeeper — and no volunteer can hold two. How many ways can she staff the day?
- Select nPr. This is an ordinary ordered arrangement, not a circle and not a derangement.
- Enter n = 10 and r = 3. Ten volunteers, three roles. The P(10,3) preset gives the same inputs in one tap.
- Leave repetition off. Nobody can hold two roles, so no volunteer is reused.
- Read the result: 720. There are 720 ways to staff the three roles.
Hand-check: 10 × 9 × 8 = 720. Change one condition and the answer changes completely: identical "helper" roles would give C(10,3) = 120, while a 3-digit code with repeats allowed gives 10³ = 1,000. Same digits, three different questions, three different answers.
Whether it is roles, rotas or seating plans, the CodexFuse permutation calculator gives you the exact count in seconds — fully managed and free to use, with no subscription and no separate app to install. When the counting is done, the rest of your IT can be just as simple.
Get Started with CodexFuseCircular permutations and derangements, without the confusion
Around a round table, rotating everyone one seat left creates no new plan — everyone keeps the same neighbours. So a circular permutation uses (n − 1)!: five people have 4! = 24 seatings, eight people 7! = 5,040. The ordinary formula would overcount every plan many times over.
A derangement is an arrangement where nothing ends up where it started: !5 = 44 ways for five letters to all reach the wrong envelopes, and !10 = 1,334,961 for ten. The same idea applies to Secret Santa draws and any reshuffle where "same as before" must be avoided.
A queue, ranking or finishing order is an ordinary permutation. Switch to Circular only when rotation genuinely repeats an arrangement, and to Derangement only when "nothing in its original place" is a hard requirement.
What mistakes make permutation answers wrong?
Most wrong answers come from three slips: counting ordered outcomes as unordered groups, allowing repeats when the problem forbids them, or treating a circle like a straight line. Check whether swapping two items changes the result, whether an item can be reused, and whether rotation creates a genuinely new arrangement before you trust any number.
- Order ignored. A combination formula for a race or ranking understates the count, sometimes sixfold.
- Repetition assumed. A PIN can reuse digits; a person cannot fill two roles. The wrong switch position silently changes the formula.
- Circle treated as a line. Counting round-table seating with n! overcounts every plan by a factor of n.
- Trusting intuition. Intuition badly underestimates counts like P(52,5). A surprisingly large number is usually the maths working, not an error.
Where permutations quietly decide real plans
A password policy is a permutation problem: a code's strength depends on how many ordered combinations an attacker must try, which is why a strong random password matters more than a memorable one. Sara's 720 assignments likewise mean any single line-up has a 1-in-720 chance at random, roughly 0.14% — a conversion a percentage calculator handles cleanly, while the factorial calculator checks the building blocks.
If your problem is about groups rather than order, read the combination calculator guide next — it shows exactly where the two formulas diverge.
What this calculator does not do
The tool gives you the number of arrangements, not a list of them — meaningless for counts like 311,875,200. It does not decide whether order matters in your problem; the swap test above is how you decide. It does not compute probabilities; it gives you the count those calculations start from. And per the tool page, use requires a free CodexFuse account sign-in, so it is not an anonymous one-click widget.
Those limits are also why the wider platform matters: most people have a dozen small tools, logins and subscriptions scattered across providers, leaking time and money. CodexFuse replaces that scatter with one fully managed platform on pay-as-you-go credits — no subscriptions, no lock-ins, no server babysitting — covering hosting, email, security and backups.
Frequently Asked Questions
What is a permutation calculator?
It counts ordered arrangements: ways to choose r items from n distinct items when order matters. P(n,r) = n! / (n − r)!, so P(10,3) = 720.
How do I calculate P(10,3)?
Multiply the top three factors of 10!: 10 × 9 × 8 = 720. In the tool, choose nPr, enter n = 10 and r = 3, or tap the P(10,3) preset.
What is the difference between a permutation and a combination?
Order. P(10,3) = 720 counts ordered trios; C(10,3) = 120 counts unordered groups, since each trio can be arranged in 3! = 6 ways.
What are circular permutations and derangements?
A circular permutation counts arrangements around a circle, where rotations repeat: (n − 1)!, so five people have 4! = 24 seatings. A derangement has nothing in its original place: !5 = 44.
Do I need an account to use the permutation calculator?
Yes — the tool page notes you sign in to use this free tool, and a free account can be created to get started. Once signed in, it is free with no subscription.
Stop counting by hand — and stop juggling separate tools
Start with the count in front of you. Open the free permutation calculator, enter n and r, and check that order, repetition and rotation match your problem. Then use the number: finalise the rota, set the password length, or lock in the seating plan with arithmetic you can defend.
Then take the same simplification to the rest of your setup. Move your hosting, business email, security and backups to CodexFuse — fully managed on pay-as-you-go credits, with no subscriptions to track and no servers to babysit. Create your free account today and put your IT on one platform that only charges for what you actually use.
