Formula
nPr = n! ÷ (n − r)!nCr = n! ÷ (r! × (n − r)!)With repetition: permutations = nʳ, combinations = C(n + r − 1, r)
- n! — n factorial, the product 1 × 2 × … × n
Use a permutation when order matters (a podium finish) and a combination when it doesn't (a hand of cards).
Worked examples
Choose 3 of 5
- Total items (n)
- 5
- Items chosen (r)
- 3
Combinations (nCr)
10
Permutations (nPr): 60
Permutations with repetition (nʳ): 125
Combinations with repetition: 35
5-card hands from 52
- Total items (n)
- 52
- Items chosen (r)
- 5
Combinations (nCr)
2,598,960
Permutations (nPr): 311,875,200
Permutations with repetition (nʳ): 380,204,032
Combinations with repetition: 3,819,816
How it works
Choosing 3 of 5 people for a podium (order matters) gives 5 × 4 × 3 = 60 permutations. Choosing 3 of 5 for a committee (order doesn't matter) divides by the 3! = 6 ways to order the same group, giving 10 combinations.
Repetition applies when the same item can be picked more than once, such as digits in a PIN: 10 digits, 4 positions gives 10⁴ = 10,000 permutations with repetition. Results are exact integers (no scientific-notation rounding) and very long answers are also shown in scientific form.
For the chance of an outcome, feed these counts into the Probability Calculator, which also handles repeated trials.
Assumptions and limits
- n and r must be whole numbers with r ≤ n and n ≤ 1,000.