Combination and Permutation Calculator
Find how many ways you can choose r items from n, as combinations (order does not matter) and permutations (order matters), with or without repetition.
Combination and permutation formulas
nCr = n! ÷ (r! × (n − r)!) nPr = n! ÷ (n − r)!
Use combinations when order does not matter, such as choosing a committee. Use permutations when order matters, such as awarding first, second and third place.
Worked example
Choosing 3 people from 10: 10C3 = 10! ÷ (3! × 7!) = 120 combinations. If the three get different roles, order matters: 10P3 = 10 × 9 × 8 = 720 permutations.
With repetition
| Type | Formula | n = 10, r = 3 |
|---|---|---|
| Combinations, no repetition | n! ÷ (r!(n − r)!) | 120 |
| Permutations, no repetition | n! ÷ (n − r)! | 720 |
| Combinations with repetition | (n + r − 1)! ÷ (r!(n − 1)!) | 220 |
| Permutations with repetition | n^r | 1,000 |
Frequently asked questions
What does nCr mean?
n choose r: the number of ways to choose r items from n when order does not matter.
What is the difference between nCr and nPr?
nPr counts ordered arrangements, so it is always at least as large as nCr. nPr = nCr × r!.
What is 0! equal to?
0! is defined as 1, so nC0 = nCn = 1.
Why are very large answers shown in scientific notation?
Factorials grow very fast, so results with more than 15 digits are shown as a number times a power of 10.
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