Permutation & Combination Calculator
Find the number of ordered arrangements (permutations) and unordered selections (combinations) of r items from n, with or without repetition. Results are exact, even when they have thousands of digits.
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How to use Permutation & Combination Calculator
- Enter n, the total number of items, and r, the number you choose.
- Tick Allow repetition if items can be used more than once.
- Read the permutation and combination counts.
- Open the steps for the formulas, and copy the full digits if the number is long.
Counting with order and without
Permutations count arrangements. Choosing a president, a secretary and a treasurer from 10 people is an ordered selection, so there are 10 × 9 × 8 = 720 ways. That is 10P3 = 10! / 7! = 720.
Combinations count groups. Picking a committee of 3 from the same 10 people ignores who holds which role, so each group is counted once: 10C3 = 720 / 3! = 120.
When repetition is allowed things change. A 2-digit code from 3 symbols with repeats has 3^2 = 9 ordered outcomes, while an unordered choice of 2 from 3 types of item has C(4, 2) = 6 outcomes. Choosing a pair from five people, with no repeats, gives 5C2 = 10 possible pairs.
Where it is used
- Probability: counting favourable outcomes divided by total outcomes.
- Passwords and PINs: the number of possible codes.
- Scheduling, seating plans and tournament pairings.
- Statistics: binomial coefficients and sampling.
- Computer science: analysing algorithms and search spaces.
Limits
n and r are limited to 10,000 to avoid slow computations, and they must be whole numbers. Results are exact integers; the headline number is abbreviated when it has more than about 22 digits, but the copy buttons give every digit. Without repetition r cannot exceed n. The calculator counts distinct items only; arrangements with identical items, such as the letters in MISSISSIPPI, need a multinomial calculation, which this tool does not do. Everything runs in your browser. As a quick check on any answer, the product of the counts follows the relationship nPr = nCr × r!, so you can verify one result from the other. Small cases are worth doing by hand first: choosing 2 of 4 items gives 12 ordered pairs but only 6 unordered pairs.
Why use this permutation & combination calculator tool
- Exact big integers. No rounding or scientific notation overflow; very large answers are summarised and can be copied in full.
- With or without repetition. Switch between the four standard counting formulas with one checkbox.
- Factorial working. See n!, r! and the formula substituted with your numbers.
- Sensible limits. Values up to 10,000 are supported, and impossible cases such as r larger than n are explained.
Frequently asked questions
What is the difference between a permutation and a combination?
In a permutation the order matters, so ABC and CBA are different. In a combination order does not matter, so they count as one selection.
What are the formulas?
nPr = n! / (n − r)! and nCr = n! / (r! (n − r)!). With repetition, permutations are n^r and combinations are C(n + r − 1, r).
Why can nCr be smaller than nPr?
Each combination corresponds to r! different orderings, so nPr = nCr × r!. Combinations are never more than permutations.
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