DIY Cost Tools
Permutation and Combination Calculator
MATH CALCULATOR
Calculate ordered permutations and unordered combinations for n distinct objects selected r at a time.
How it works
Permutations count selections where order matters. Combinations count groups where rearranging the selected items does not create a new outcome.
Formulas
nPr = n! ÷ (n − r)!
nCr = n! ÷ [r!(n − r)!]
Example
Choosing 3 items from 10 gives 720 ordered permutations but only 120 unordered combinations. Each three-item group has 3! = 6 possible arrangements.
Important limits
- The calculator assumes distinct objects and selection without replacement.
- Inputs are capped at 200 for responsive exact-integer output.
- Repeated objects or sampling with replacement require different formulas.
Frequently asked questions
How do I know whether order matters?
If ABC and BAC count as different outcomes, use permutations. If they represent the same group, use combinations.
What happens when r is zero?
There is one way to select or arrange nothing, so both results equal 1.
Permutation or combination?
Use a permutation when order matters and a combination when it does not. Choosing a president, vice president, and treasurer is a permutation because the roles differ; choosing three committee members is a combination because the selected group is what matters.
Standard formulas usually assume items are distinct and selected without replacement. If repetition is allowed, or if some items are identical, a different counting rule may be required. Check that r does not exceed n for selections without replacement. Factorials grow very quickly, so large inputs may produce enormous results even when the setup is simple. Writing out a small example can help confirm whether changing the order creates a new outcome.
Review notes
Results use exact BigInt arithmetic and standard counting formulas. Last reviewed September 30, 2026.