Calculation details
Planning estimate only, not financial, tax, or legal advice. Verify assumptions and current rules before making decisions.
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Use this result
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What does this calculator estimate?
A permutations calculator counts ordered arrangements. Enter n and r to see how many ways to arrange r of n items.
- nPr = n! ÷ (n−r)!
- 5P3 = 60
- Order matters (vs combinations)
What permutations count
Permutations count ordered arrangements — 3 runners in 5 lanes, password orders, or top-3 finishes. The formula divides the factorial by the leftover factorial, removing the unselected items. Order matters: ABC differs from CBA.
Limitations to watch for
The formula assumes distinct items without repetition. Large n and r produce enormous numbers — the tool handles the math within practical limits. If order doesn't matter, use combinations instead.
How to use it in practice
Use nPr for arrangements, rankings, and sequences. Compare with nCr: 5P3 = 60 (ordered) vs 5C3 = 10 (unordered). For repetitions allowed, the count is n^r instead.
['Enter the total items n.', 'Enter the arranged size r.', 'Read the number of permutations.']
How this calculator works
Formula
nPr = n! ÷ (n − r)! — the number of ways to arrange r items from n when order matters.
Worked example
Arranging 3 items from 5: 5P3 = 60 permutations.
Assumptions to verify
- Items are distinct.
- No repetition is allowed.
- Order matters.
Frequently asked questions
What is a permutation?
An ordered arrangement: 5P3 = 60 ways to arrange 3 of 5 items.
How is nPr calculated?
n! ÷ (n−r)! — total arrangements minus the unselected items.
What's the difference from combinations?
Order matters in permutations (ABC ≠ CBA); it doesn't in combinations.
What are permutations used for?
Rankings, arrangements, passwords, and any ordered selection.
Can items repeat?
No — the standard formula assumes distinct items. With repetition, use n^r.
Why do permutations explode so fast?
Each slot multiplies the remaining options — 10P5 is already 30,240.
How many ways can 3 runners finish?
If 5 runners compete for 3 medal spots: 5P3 = 60 possible medal orders.
Cite this tool
BoringToolsKit. “Permutations Calculator (nPr).” boringtoolskit.com/permutations-calculator/ (reviewed 2026-08-25). Free to reference in articles, syllabi, and answer posts with a link.
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