Permutation and combination calculator
Count arrangements (permutations) and selections (combinations) of r items from n. Results are exact even for large numbers, and cases with repetition are included.
Worked example
With the values shown above when the page opens:
10C3 = 120, 10P3 = 720
- nPr = n × (n − 1) × ... × (n − r + 1) = n! ÷ (n − r)!
- nCr = nPr ÷ r! = n! ÷ (r! × (n − r)!)
- With repetition allowed: arrangements = n^r, combinations = C(n + r − 1, r)
How it works
- A permutation counts ordered arrangements: choosing 3 of 10 people for gold, silver and bronze gives 10 × 9 × 8 = 720.
- A combination counts selections where order does not matter: choosing 3 of 10 for a committee gives 720 ÷ 3! = 120.
- With repetition allowed (the same item can be used again) arrangements are n^r, and combinations are C(n + r − 1, r).
- Every combination of r items can be arranged in r! ways, which is why nPr = nCr × r!.
Questions
How do I know whether to use a permutation or a combination?
Ask whether swapping two of the chosen items gives a different outcome. If it does (a race result), use a permutation. If not (a team), use a combination.
What is 0! ?
0! is defined as 1, which makes the formulas work. For example, nC0 = 1: there is exactly one way to choose nothing.
How big can the numbers get?
Very big. 52C5 (poker hands) is 2,598,960, and 100C50 has 30 digits. This calculator is exact for n up to 1000.
Related
Calculators are for learning and quick checks. Check important figures yourself, and do not rely on them for tax, legal or financial decisions.