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Permutation & Combination Calculator
Count ordered permutations (nPr) and unordered combinations (nCr) for choosing r items from n distinct items.
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Transparent methodology
What the result includes—and what it does not
Built for students, teachers, and anyone counting how many ways r items can be selected from n distinct items — once where the order of selection matters, and once where it does not.
Calculation method
- 1Both inputs are truncated to whole numbers, and the calculation runs only when 0 ≤ r ≤ n.
- 2The permutation count is built as the falling product n × (n−1) × … × (n−r+1), which is nPr = n! / (n−r)!.
- 3The combination count divides that permutation total by r! — the number of ways the same r items can be ordered — giving nCr = n! / (r! × (n−r)!).
- 4Both counts treat the n items as distinct and allow each item to be used at most once, so nCr is the number of combinations without repetition.
Assumptions and limits
- Repetition is not counted: an item picked once cannot be picked again. Permutations with repetition (n^r) are not calculated here at all, and combinations with repetition need the substitution described in the FAQ.
- The tool counts arrangements; it does not list or generate them. It takes only the two numbers n and r, never the items themselves.
- Both fields cap at 170, and results beyond 2^53 (9,007,199,254,740,992) are rounded by double-precision arithmetic rather than exact.
- Entering r greater than n returns an input error rather than the mathematical answer of 0 arrangements.
Worked example
Choosing 3 items from 10
Input: n = 10 total items and r = 3 chosen — the tool's default values.
Output: 720 permutations (nPr) and 120 combinations (nCr) — nPr is 10 × 9 × 8 = 720, and dividing by 3! = 6 gives nCr = 120.
Primary sources
- NIST Digital Library of Mathematical Functions — §26.3 Binomial CoefficientsDefines the binomial coefficient as the number of ways of choosing n objects from m distinct objects without regard to order, with the factorial formula m! / ((m−n)! n!).
- MIT OpenCourseWare — Mathematics for Computer Science (6.042J)Free university course whose counting unit covers permutations, combinations, and the counting principles behind nPr and nCr.
How to use it
Three steps. One clear answer.
- 1Enter or paste the values requested by the tool.
- 2Review the input units and choose any relevant options.
- 3Calculate, then copy or download the result you need.
Built for trust
Useful without the friction.
- No registration or paywall before your result.
- Responsive controls for phone, tablet, and desktop.
- Clear assumptions and warnings where estimates have limits.
Frequently asked questions
What is the difference between a permutation and a combination?
A permutation counts ordered selections, so ABC and CAB are two different results: nPr = n! / (n−r)!. A combination counts unordered selections, so ABC and CAB are the same result: nCr = n! / (r! × (n−r)!). Dividing nPr by r! is exactly what removes the ordering.
How do I calculate combinations without repetition?
That is the nCr figure this calculator returns by default — each item can be chosen at most once. Enter the size of the set as n and how many you are picking as r; with n = 10 and r = 3 the answer is 120.
How many possible combinations are there from a set of items?
It depends on how many you draw at a time: enter the set size as n and the draw size as r. If you want the number of subsets of every size, that total is 2^n, which this tool does not compute in one step.
Can this handle combinations or permutations with repetition?
Not directly. Permutations with repetition are simply n^r, plain arithmetic you can do elsewhere. For combinations with repetition, use the identity C(n+r−1, r): to pick 3 from 10 types with repeats allowed, enter n = 12 and r = 3 to get 220.
Does it generate the actual list of permutations?
No. It returns the counts only — it has no field for the items themselves, so it cannot print or export the individual arrangements.
Can I use this to build keyword combinations for SEO?
No, this page is pure combinatorics on the numbers n and r. To join modifiers, topics, and locations into an actual keyword list, use the Keyword Combination Generator instead.
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