Permutation and Combination Calculator
Enter the total number of items n and how many are chosen r. The calculator gives the number of ordered arrangements and the number of unordered selections.
Result
- Permutations with repetition (nʳ)
- 1,000
- Combinations with repetition
- 220
- n!
- 3,628,800
Formula
- nPr = n! ÷ (n − r)!
- nCr = n! ÷ (r! × (n − r)!)
- With repetition: permutations = nʳ, combinations = (n + r − 1)! ÷ (r! × (n − 1)!)
How the Permutation and Combination Calculator works
A permutation is an arrangement where order matters, like the finishing order of a race. A combination is a selection where order does not matter, like the members of a committee.
There are always at least as many permutations as combinations, because each combination of r items can be arranged in r! different orders. Dividing the permutations by r! gives the combinations.
Worked example
Choosing 3 items from 10: permutations = 10 × 9 × 8 = 720. Each group of 3 can be ordered 3! = 6 ways, so combinations = 720 ÷ 6 = 120.
Frequently asked questions
When do I use a permutation instead of a combination?
Use a permutation when the order of the chosen items matters, such as a password or a ranking. Use a combination when only the membership matters, such as a hand of cards.
What is 0 factorial?
0! is defined as 1. There is exactly one way to arrange nothing, and the definition keeps the formulas consistent.
How many ways can I choose 6 numbers from 49?
49C6 = 13,983,816, which is why the odds of matching all six in such a lottery are about 1 in 14 million.
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