Find nPr — permutations of n items taken r at a time — with or without repetition, computed exactly using BigInt arithmetic.
Enter n and r and click Calculate to see the result.
This free permutation calculator finds nPr — the number of ways to arrange r items chosen from a set of n, where the order of arrangement matters — with exact results computed using BigInt arithmetic, so even large factorials never lose precision to floating-point rounding. It also covers permutations with repetition allowed, where the same item can be reused across positions. Whether you're solving a probability homework problem, estimating a password's keyspace, or counting race-finish orderings, this permutation calculator with steps shows the exact factorial substitution behind every answer.
A permutation counts ordered arrangements: choosing r items out of n and arranging them in a specific sequence, where swapping the order of any two chosen items creates a distinct outcome. This calculator computes the "without repetition" case (nPr = n!/(n−r)!, the classic permutation formula) and the "with repetition" case (n^r, where items can repeat across positions).
Students and teachers working through combinatorics and probability coursework, computer scientists estimating password or key-space sizes, statisticians computing arrangement probabilities, event organizers counting seating or scheduling orders, and anyone who needs an exact — not approximate — count of ordered arrangements can use this tool.
Many real problems hinge on whether order matters. A race's first-second-third finish order, a PIN code, a book arrangement on a shelf, or a relay-team running order are all permutation problems, not combination problems, because rearranging the same items produces a genuinely different outcome. Getting this distinction right is essential for correct probability calculations.
Cryptographers estimate password and key-space sizes using permutation counts. Race organizers calculate the number of possible podium orderings. Schedulers count the ways to sequence tasks or appointments. Geneticists count possible DNA/RNA sequence orderings. Statisticians use permutations as the building block for many probability distribution formulas.
How this calculator counts ordered arrangements
Permutations treat different orderings of the same items as distinct outcomes — ABC and BCA count as two separate permutations.
Without repetition, each choice shrinks the pool by one; with repetition, every position has the full n choices again, giving the much larger n^r.
When you arrange every item (r=n), the permutation formula reduces to the plain factorial, since (n−n)! = 0! = 1.
From choosing a mode to reading the exact count
Pick "Without Repetition" (the standard nPr case) or "With Repetition" depending on your problem.
Type the total number of items (n) and how many you're arranging (r).
The calculator applies the matching permutation formula instantly using exact BigInt arithmetic.
See the exact permutation count, plus the factorial substitution used to reach it.
For small n and r, manually list a few arrangements to spot-check the calculator's count.
Arranging 3 runners on a podium out of 10 racers
10 runners finish a race. In how many distinct orders can 3 of them fill 1st, 2nd, and 3rd place?
Explanation: There are 720 distinct ways to fill the podium, because swapping which runner takes 1st vs 2nd vs 3rd creates a genuinely different outcome. If repetition were somehow allowed (it isn't, in a real race), the count would instead be 10³ = 1000.
What the result represents
| Mode | What the Result Means |
|---|---|
| Without Repetition | The exact count of ordered arrangements of r items chosen from n, with no item reused |
| With Repetition | The exact count of ordered arrangements of r positions where any of the n items can be reused |
Typical ranges: permutation counts grow extremely fast (factorially or exponentially) as n and r increase — even modest values like n=20, r=10 produce results in the billions.
Manual verification: for small n and r, list out a handful of arrangements by hand and confirm the pattern matches the displayed factorial substitution.
Where counting ordered arrangements matters
Check permutation word problems step by step, with the exact factorial shown.
Count the number of possible podium or finishing-order outcomes.
Estimate a keyspace size using permutations with repetition.
Analyze the complexity of algorithms that enumerate orderings.
Count possible sequence orderings in DNA/RNA analysis.
Count the ways to sequence a fixed set of tasks or appointments.
Work out how many ways guests can be seated in a row of chairs.
Count possible card, tile, or piece orderings in games and puzzles.
Use permutation counts as building blocks for probability calculations.
Practice the permutation problems common in SAT/GRE/GMAT quantitative sections.
Analyze ordered symbol sequences in signal and coding theory.
Count possible ordered outcomes for multi-round competitions.
What this permutation calculator does well, and where it doesn't apply
The single question that tells them apart: does order matter?
| Concept | Order Matters? | Formula | Example (n=10, r=3) |
|---|---|---|---|
| Permutation (nPr) | Yes | n!/(n−r)! | 720 |
| Combination (nCr) | No | n!/(r!(n−r)!) | 120 |
Summary: This permutation calculator gives you an instant, free, exact way to count ordered arrangements — with or without repetition — using BigInt-precision factorial math. Pair it with the Combination Calculator when order doesn't matter, and the Factorial Calculator to check the underlying factorial values.
Common questions about permutations
Trusted educational references to go deeper on combinatorics
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