Permutation and Combination Calculator
Enter the total number of items n and how many you choose, r. Pick whether items can repeat. You get the number of ordered arrangements (permutations) and the number of unordered selections (combinations).
Your numbers
How we calculate it
Without repetition, the number of ordered arrangements is nPr = n! ÷ (n − r)!, and the number of selections where order does not matter is nCr = n! ÷ (r! × (n − r)!). With 10 items and 3 chosen, nPr is 720 and nCr is 120.
With repetition allowed, the ordered arrangements number n^r, and the unordered selections number (n + r − 1)! ÷ (r! × (n − 1)!). With 10 items and 3 chosen that is 1,000 and 220. The arithmetic is exact. Results above 15 digits are shown in scientific notation.
Assumptions and limits
- Without repetition, r cannot be larger than n. With repetition, n must be at least 1.
- n and r can each be up to 170.
- Permutations count order (ABC and CBA are different) and combinations ignore it.
Frequently asked questions
What is the difference between a permutation and a combination?
In a permutation the order matters, so choosing A then B is different from B then A. In a combination only the group matters. Picking a team is a combination, while ranking the top three is a permutation.
How many ways can I choose 5 cards from 52?
The number of 5 card hands is 52 choose 5, which is 2,598,960.
What does repetition allowed mean?
It means the same item can be chosen more than once, such as the digits of a code where each digit can repeat. A 4 digit code from 10 digits has 10^4 = 10,000 arrangements.
Related calculators
Worked example (a fixed example using the starting numbers, it does not change as you type)
Inputs: Total number of items (n) = 10; Number chosen (r) = 3; Repetition = Not allowed.
- Permutations (order matters) 720
- Combinations (order does not matter) 120
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