Free Permutation & Combination Calculator Online

A permutation & combination calculator answers a question that trips up almost everyone the first time they meet it in a statistics or algebra class: does the order of your selection matter, and if it doesn't, how many ways can you still make it? This tool takes a set size (n) and a chosen count (r) and instantly returns both the permutation P(n,r) and the combination C(n,r), along with the repetition-allowed variants of each, so you never have to remember which factorial formula goes with which scenario. Below is exactly how the calculator works, when you'd reach for it instead of a factorial table, and the honest math behind why these numbers grow the way they do.

How the Permutation & Combination Calculator Works

The interface is deliberately narrow: two number fields, n for the total items in your set and r for how many you're choosing from it. As soon as both fields hold valid non-negative integers with r no greater than n, four result cards update instantly — there's no submit button and no page reload, since the calculation runs live in your browser the moment you stop typing.

• Permutation P(n,r), computed as n!/(n−r)!, for when the order of the selected items matters — think of ranking three finishers out of eight racers.
• Combination C(n,r), computed as n!/(r!(n−r)!), for when order is irrelevant — picking a 5-person committee from a pool of 20 doesn't care who was chosen first.
• Permutations with repetition, computed as n^r, for situations where the same item can be selected more than once, such as counting possible 4-digit PIN codes.
• Combinations with repetition, computed as (n+r−1)!/(r!(n−1)!), the less common but still useful case of choosing r items from n types where repeats are allowed and order doesn't matter.

Each result card shows the formula it used directly underneath the number, so you can check your own hand calculation step against the tool's output rather than just trusting a black-box answer. Very large results switch automatically to exponential notation once they cross roughly a quadrillion, keeping the display readable instead of overflowing into an unreadable string of digits.

If you need to run this same nPr/nCr calculation from inside a script, spreadsheet macro, or backend service rather than through the browser form, the identical logic is exposed as a metered REST endpoint at /docs, billed at one credit per call and built on exact integer arithmetic so results for large n stay precise instead of drifting the way floating-point factorials eventually do.

Why Use a Permutation & Combination Calculator

Order-versus-no-order counting shows up constantly outside the classroom, and getting the formula wrong usually means getting the answer wrong by an order of magnitude:

• Probability problems. Working out the odds of a specific poker hand, lottery draw, or raffle outcome almost always starts with a combination count for the total possible outcomes, then a second combination count for the favorable ones.
• Scheduling and arrangements. Assigning a fixed order to a subset of items — who presents first, second, and third out of a group of ten, or how many distinct finishing orders are possible in an eight-runner race — is a straightforward permutation.
• Team and committee selection. Choosing an unordered group, like a 4-person project team from 12 candidates or a jury panel from a larger pool, is the classic combination scenario, and the numbers get large fast even with modest pool sizes.
• Password and code space estimates. Counting how many distinct 6-character codes exist from a 36-character alphabet, where the same character can repeat, is exactly the permutation-with-repetition case (36^6), useful for a rough sense of brute-force difficulty.
• Statistics and combinatorics coursework. Binomial coefficients, expected value calculations, and hypergeometric distribution problems all lean on nCr as a building block, so a fast, verifiable calculator saves real time across a problem set.

Permutations vs Combinations: The Technical Difference

The entire distinction comes down to one question: if you rearrange the selected items, do you get a different result? Ranking three medalists out of ten athletes is a permutation — gold, silver, and bronze going to Alice, Bob, and Carla is a different outcome than the same three people in a different order. Selecting three athletes to send to a different event, with no ranking involved, is a combination — the same three names count once no matter what order they were picked in.

Mathematically, a combination is a permutation divided by r! — the number of ways to reorder the r chosen items. That's why C(n,r) is always P(n,r)/r!, and why C(n,r) is always smaller than or equal to P(n,r) for the same n and r. When r equals 0 or r equals n, both values collapse to either 1 or n!, since there's either nothing left to arrange or everything is being arranged.

Factorial growth itself is worth understanding, because it's the reason these numbers escalate so fast. 10! is 3,628,800. 15! is already past a trillion. 20! is roughly 2.43 × 10^18, larger than the number of seconds since the universe began. Standard double-precision floating point, the number format most calculators and spreadsheet programs use internally, loses the ability to represent factorials exactly somewhere past 170! — beyond that point the value overflows to infinity rather than returning a real (if enormous) number. That's precisely why an API built for combinatorics at scale needs exact integer arithmetic rather than floating point: this tool's underlying engine uses arbitrary-precision integers for its calculations, so a result like 1000P1000 — a number with over 2,500 digits — comes back exact rather than rounded or overflowed.

Scenario	Formula	Order matters?	Repeats allowed?
Ranking finishers	P(n,r) = n!/(n−r)!	Yes	No
Choosing a committee	C(n,r) = n!/(r!(n−r)!)	No	No
PIN codes, license plates	n^r	Yes	Yes
Multisets, repeated-choice groupings	(n+r−1)!/(r!(n−1)!)	No	Yes

Manual computation is where most errors creep in — dropping a factorial term, forgetting to divide by r!, or simply mistyping a multiplication chain that's 15 terms long. A calculator that shows its formula alongside the number gives you a way to catch that kind of mistake immediately rather than after it's already propagated into a wrong probability estimate.

Permutation & Combination Calculator vs Other Methods

There's more than one way to get an nPr or nCr value, and each has a real tradeoff worth knowing:

• Scientific calculators (physical or phone app). Most have dedicated nPr and nCr buttons, which is convenient, but the vast majority run on the same double-precision floating point that overflows past roughly 170!, so very large n values return an error or a nonsensical result rather than a correct one.
• Spreadsheet formulas. Excel and Google Sheets both offer PERMUT and COMBIN functions, which work well for everyday values but still inherit the same floating-point ceiling and require opening a spreadsheet just to check one number.
• Manual factorial tables or long multiplication. Reliable for small n, but the arithmetic gets unwieldy past n=10 or so, and a single dropped digit anywhere in the chain invalidates the whole result.
• Computational engines like Wolfram Alpha. Excellent when you need a full symbolic derivation or a multi-step probability walkthrough, but overkill and slower when all you need is a direct number for a specific n and r.
• This calculator. Built specifically for the direct lookup case — type two numbers, get all four counting variants at once with their formulas shown, no software installation and no floating-point ceiling on the API side.

A Worked Example: P(8,3) and C(8,3)

Seeing the arithmetic laid out once makes the formulas easier to trust. Take n=8 and r=3 — say, choosing from eight finalists for three distinct prizes versus choosing an unordered group of three from the same eight. For the permutation, P(8,3) = 8!/(8−3)! = 8!/5!, which cancels down to 8 × 7 × 6 = 336 distinct ordered outcomes. For the combination, C(8,3) = 8!/(3!×5!) = 336/6 = 56, since each unordered group of three corresponds to 3! = 6 different orderings that the permutation count treats as separate. That relationship — combination equals permutation divided by r! — holds for every valid n and r, and it's a fast way to sanity-check either number by hand once you have the other.

Repetition changes the picture further. If those same three prize slots could each go to any of the eight finalists again, including repeats, the permutation-with-repetition count jumps to 8³ = 512, larger than the no-repeats permutation because reusing a finalist opens up combinations that were previously off-limits. Running all four variants side by side on the same n and r is exactly what makes it easy to see how much a single assumption — order, repetition — changes the final count.

How Your Numbers Are Processed

Everything happens inside your browser tab. The n and r values you type are read by JavaScript running locally and run through the factorial-based formulas, with the four results rendered straight back into the page. Nothing about what you typed travels to a server, gets logged, or persists in any account, since there's no account tied to the calculator in the first place. Close the tab and the session is gone, exactly as if it never happened.

Common Questions About Permutations and Combinations

What's the actual practical difference between a permutation and a combination?

It comes down to whether rearranging your selection changes the outcome. If ordering the same group differently counts as a new result — like a race podium — you want a permutation. If the group is the outcome regardless of order — like a hand of cards you're holding — you want a combination.

Why do these numbers grow so much faster than I expect?

Because factorials multiply together every integer up to n, growth is closer to exponential-times-polynomial than linear. Going from n=10 to n=15 doesn't multiply the result by 1.5 — it multiplies it by roughly 15!/10!, which is over 360,000. Small increases in n produce enormous jumps in the count.

Can this calculator handle repetition, like counting PIN codes or license plates?

Yes — the permutations-with-repetition card (n^r) and the combinations-with-repetition card ((n+r−1)!/(r!(n−1)!)) cover exactly that case, alongside the standard no-repeats versions, so you get all four counting scenarios from the same two inputs.

What does "n choose r" actually mean?

"n choose r" is spoken shorthand for the combination C(n,r) — the number of distinct unordered groups of size r you can form from a pool of n items. It's the same value shown on the Combination card in this tool, and it's the building block behind binomial coefficients in algebra and statistics.

Why did my phone's calculator give an error or a strange answer for a large n?

Most consumer calculators use standard floating-point math internally, which can't represent factorials accurately once n climbs into the high hundreds — the true value overflows the format entirely. This tool's API layer avoids that by using exact integer arithmetic instead of floating point for its computations.

Is a permutation and combination calculator only useful for homework?

No — it's just as relevant for estimating password search spaces, structuring probability calculations for games and raffles, or validating combinatorics logic inside an application through the API, where getting nPr or nCr wrong would silently produce an incorrect probability or count somewhere downstream.

Related Tools

If you're working through a probability problem that starts with a permutation or combination count, the Probability Calculator picks up right where this tool leaves off, turning a favorable-outcomes-over-total-outcomes ratio into a clean probability. For the arithmetic that often surrounds these problems — square roots, exponents, and general number crunching — the Scientific Calculator is a faster stop than switching tools entirely. If your combinatorics work feeds into a statistics assignment, Standard Deviation Calculator and Mean, Median & Mode Calculator cover the descriptive-statistics side of the same coursework. And when a combinatorics result needs to be expressed as a clean fraction rather than a decimal probability, the Fraction Calculator handles the simplification.