Permutation & Combination Calculator
Calculate nPr (permutations) and nCr (combinations) with a step-by-step formula breakdown. Enter n and r to see the count, formula expansion, and intermediate values.
Permutations (nPr)
720
Combinations (nCr)
120
How to use the permutation & combination calculator
Enter total items n and the selection size r (0 ≤ r ≤ n).
Formula & explanation
Permutations: nPr = n! ÷ (n−r)! (order matters). Combinations: nCr = n! ÷ (r! × (n−r)!) (order doesn't).
Examples
10P3 = 720, 10C3 = 120. 5C2 = 10 (the number of pairs in a group of 5).
Frequently asked questions
- What is the difference between permutations and combinations?
- Permutations count arrangements where order matters (e.g. PIN codes, race placements). Combinations count selections where order doesn't matter (e.g. lottery numbers, team selection).
- Why is nPr always ≥ nCr?
- Because nPr = nCr × r!. Every combination can be arranged in r! ways, so permutations are always a multiple of combinations for the same n and r.
- What does r = 0 or r = n mean?
- nC0 = 1 (there is exactly one way to choose nothing). nCn = 1 (there is exactly one way to choose everything). nP0 = 1 and nPn = n! (all arrangements).
- What is the maximum n this calculator supports?
- Up to n = 200, using BigInt arithmetic to handle the very large factorials that arise for large n values.
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