Combinations and Permutations Calculator
Counting how many ways you can choose or arrange items is the foundation of probability. Enter the total number of items n and how many you select r to see all four counts at once: permutations and combinations, each with and without repetition. Permutations care about order, combinations do not.
Counting formulas
Permutations, no repeats: nPr = n! / (n - r)!
Combinations, no repeats: nCr = n! / (r! * (n - r)!)
Permutations with repeats: n^r
Combinations with repeats: (n + r - 1)! / (r! * (n - 1)!)
Order matters for permutations and not for combinations. Repetition allows the same item to be chosen more than once. Without repetition, r must not exceed n.
Worked example
From 10 items choosing 3: combinations without repeats is 10! divided by (3! times 7!) = 120. Permutations without repeats is 10! divided by 7! = 720. Permutations with repeats is 10^3 = 1,000. Combinations with repeats is 12! divided by (3! times 9!) = 220.
Combinations and permutations: frequently asked questions
What is the difference between a combination and a permutation?
A permutation counts ordered arrangements, where the order matters, such as ranking finishers in a race. A combination counts unordered selections, where order does not matter, such as choosing a committee. For the same n and r, there are always at least as many permutations as combinations.
What are the formulas?
Permutations without repetition: nPr = n! divided by (n - r)!. Combinations without repetition: nCr = n! divided by (r! times (n - r)!). With repetition, permutations are n^r and combinations are (n + r - 1)! divided by (r! times (n - 1)!).
When is repetition allowed?
Repetition is allowed when an item can be chosen more than once, such as a PIN where digits can repeat. It is not allowed when each item is used at most once, such as dealing distinct cards. This calculator shows both cases side by side.
What if r is greater than n?
Without repetition, choosing more items than exist is impossible, so the count is zero. With repetition, r can exceed n because items repeat. The calculator handles both and returns zero where a selection is impossible.
Sources
- NIST Digital Library of Mathematical Functions: Combinatorial analysis, binomial coefficients.
Reviewed by the CalculatorHub team, edited by James Graham, 19 June 2026. See our methodology.