Statistics / Counting
Permutation and Combination Calculator (nPr, nCr)
Count the ways to choose r items from n, as ordered permutations (nPr) and unordered combinations (nCr), with or without repetition, as exact whole numbers.
Permutation and Combination Calculator (nPr, nCr): Permutations count ordered choices: nPr = n! ÷ (n − r)!; picking first, second, and third place from 10 runners gives 10 × 9 × 8 = 720. Combinations ignore order: nCr = n! ÷ (r! (n − r)!); any 3 of 10 gives 720 ÷ 6 = 120. With repetition, ordered choices number nʳ (a 4-digit PIN has 10⁴ = 10,000) and unordered ones C(n + r − 1, r). The arithmetic is exact, even for very large results. Runs 100% locally in your browser with zero server file uploads.
- Category
- School & study tools
- Runs
- In your browser
- Cost
- Free · no sign-up
- Availability
- Ready to use
Runs entirely in your browser
Permutations count ordered arrangements: nPr = n! ÷ (n − r)!, so choosing a gold, silver, and bronze winner from 10 runners gives 720. Combinations ignore order: nCr = n! ÷ (r! (n − r)!), so choosing any 3 of 10 gives 120. With repetition allowed, ordered choices number nʳ and unordered ones C(n + r − 1, r). Results are exact; very large ones are shown in scientific notation.
Worked examples
A committee of 4 from 12 people: C(12, 4) = 495. A chair, secretary, and treasurer from 12: P(12, 3) = 1,320. Arrangements of a 52-card deck: 52! ≈ 8.07 × 10⁶⁷.
To pick items at random instead of counting them, use the random number generator.
Factorials
n! multiplies all whole numbers from 1 to n: 5! = 120, 10! = 3,628,800, and 0! = 1 by definition, so that C(n, 0) = C(n, n) = 1.
Combinations appear as the coefficients of (a + b)ⁿ and in Pascal's triangle, where each number is the sum of the two above it.
How to use it
- Enter the total number of items, n.
- Enter how many are chosen, r, and tick if items can repeat.
- Read the number of permutations and combinations.
Privacy & limitations
Everything is calculated in your browser.
Related tools
Frequently asked questions
How do I know if order matters?
If swapping two chosen items gives a different outcome, such as positions in a race or digits in a code, use permutations; if not, such as a hand of cards or a committee, use combinations.
What are the odds of winning a 6 from 49 lottery?
There are C(49, 6) = 13,983,816 combinations, so one ticket has a 1 in 13,983,816 chance of the jackpot.
Why is a combination lock really a permutation?
Because the order of the numbers matters and they can repeat: a 3-wheel lock with digits 0–9 has 10³ = 1,000 settings.
Free tool · runs in your browser · no account required