Student solving an ordered-selection problem
Know whether order changes the outcome.
Choose without repetition and enter available items and ordered positions.
Gets the exact P(n,r) count and does not confuse it with a combination.
Permutation Calculator
Count ordered arrangements exactly with or without repetition using arbitrary-precision integers and explicit model boundaries.
Specialist tool for Math Science1 documented formula1 guided projectPrivate in your browser
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A quick decision brief for this specific tool
Inputs that matter
Whole available-item or symbol count n, whole ordered-position count r, and an explicit without- or with-repetition model
Output to expect
Exact arbitrary-precision ordered-arrangement count, digit count, and the actual P(n,r) or nʳ formula used
How it works
Multiplies the r descending factors without repetition or raises n symbol choices to r ordered positions when repetition is allowed, all with BigInt
- Use permutations only when changing order creates a different outcome; use combinations when the same selected items in a different order remain one outcome.
- Without repetition r cannot exceed n; with repetition a nonempty sequence needs at least one symbol type. Confirm any application-specific exclusions separately.
Choose your path
Built around the job you need to finish
Count ordered arrangements exactly, with an explicit distinction between distinct items without reuse and repeatable symbol types.
Developer sizing a code space
Count fixed-length ordered strings when symbols may repeat.
Choose with repetition and enter symbol types and positions.
Gets exact nʳ output without Number overflow or rounding.
Analyst checking a large search space
Retain every digit and understand the bound.
Enter valid whole counts through 500 and inspect the exact integer and digit count.
Can copy an arbitrary-precision result and sees invalid cases rejected.
Authoritative checks for this tool
Outputs and checklists are planning aids. Review the linked current authorities and the records, terms, instructions, and requirements that apply to your exact situation before a consequential decision.
- Contemporary Mathematics — PermutationsOpenStax, Rice University · Educational definition of permutations as ordered lists, including the distinction from unordered combinations.
- College Algebra 2e — Counting PrinciplesOpenStax, Rice University · Permutation formula, multiplication principle, and ordered-selection examples with and without reuse.
- ECMAScript — BigInt ObjectsEcma International · Normative arbitrary-precision integer type used so large ordered counts do not overflow or silently round.
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Projects using Permutation Calculator
This tool is part of these guided projects. Each project provides step-by-step instructions with checklists and all the tools you need in one place.
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Reference & details
How it works
Updated August 2026
How it works
Updated August 2026Without repetition
For n distinct items placed into r ordered positions without reuse, multiply the r descending factors from n.
P(n,r) = n!/(n−r)!With repetition
When each ordered position independently allows any of n symbol types, multiply n choices across r positions.
nʳExact bounded arithmetic
Whole counts from 0 through 500 are evaluated with BigInt. Impossible nonrepeating selections and nonempty repeated arrangements with zero symbols are rejected.
Updated: August 2026
Example Scenarios
A student counts ways to assign three distinct offices from ten people because swapping roles changes the outcome.
A developer counts fixed-length ordered strings when every position can reuse any available symbol type.
An analyst retains the complete exact integer and digit count for a large but bounded arrangement model.
Common Mistakes to Avoid
Common Mistakes to Avoid
Using a permutation when order does not matter
Use a combination when different orders of the same selected items represent one outcome.
Treating repeatable types as distinct one-use items
Select the repetition model that matches whether each symbol type can occupy more than one position.
FAQ
A permutation counts ordered outcomes, so changing positions creates a different result. A combination ignores the order of the selected items.
Without repetition, n is the number of distinct available items. With repetition, n is the number of symbol types available at each ordered position.
A distinct item can occupy at most one position, so a nonrepeating arrangement cannot use more positions than available items.
Yes. The engine uses BigInt for every accepted case, so the displayed integer is not rounded or converted through floating-point factorial arithmetic.
It counts one empty arrangement. This identity keeps the multiplication and counting formulas consistent.
About Permutation Calculator
Choose whether symbol types may repeat, then enter n and r. The calculator returns the complete exact integer and keeps ordered arrangements distinct from unordered combinations.