Permutation and Combination Calculator
Count ordered or unordered selections, with or without repetition. Read the exact integer result, compare all four cases and list examples for small problems.
YOUR CALCULATION
Result
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Calculation steps
Work through one step at a time or reveal the complete calculation. The steps use your current inputs.
How to calculate
Without repetition, ordered selections use n!/(n−r)! and unordered selections use n!/[r!(n−r)!]. Repetition gives nʳ ordered sequences, or C(n+r−1,r) unordered multisets. The calculation uses exact integer arithmetic.
No repetition: P(n,r)=n!/(n−r)!, C(n,r)=n!/[r!(n−r)!]; repetition: nʳ or C(n+r−1,r)
Worked example
Selecting 3 distinct choices from 5 gives 10 unordered combinations or 60 ordered selections. Allowing repetition gives 35 unordered multisets or 125 ordered sequences.
Assumptions and input details
The n choices are distinct categories; repeated selections mean choosing the same category again, not arranging a fixed inventory containing duplicates. Selecting zero objects has one empty selection, including when n=0. Without repetition, selecting more than n gives zero. Counts are not probabilities. Examples are fully listed only when n≤8, r≤6 and the total is at most 60; otherwise only counts are shown.
References: OpenStax: permutations · OpenStax: combinations
Using this calculator
Can I enter my own problem?
Yes. Replace the example values with your own inputs. The result, calculation steps, tables and applicable diagrams update from those inputs. Blank or invalid values clear the result; example values are never silently substituted for missing inputs.
Are the results exact?
Fraction and counting results use exact integer arithmetic. Any decimal or percentage display is approximate.
Is my input uploaded?
The calculation runs in your browser. No account, file upload or external AI service is required. Inputs are not saved between page reloads.
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