Permutation and Combination Calculator
Calculate nPr and nCr.
Tip: Permutation: order matters (ABC ≠ BAC). Combination: it doesn't (ABC = BAC).
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About this digital tool
The permutation and combination calculator finds how many ways there are to choose r objects out of n objects in total. Enter n and r and get at once the permutations (nPr), where order matters, and the combinations (nCr), where order does not matter. Make sure r is not larger than n.
The formulas are nPr = n! ÷ (n − r)! and nCr = n! ÷ (r! × (n − r)!). For n = 5 and r = 3 you get 5P3 = 60 and 5C3 = 10. In a lottery of 6 numbers out of 49 the combinations are 13,983,816, so the chance of winning with one ticket is about 1 in 14 million. It is used in probability, in games and in organising teams.
How to use
- Enter n, the number of all available objects.
- Enter r, the number of objects you choose (r must not exceed n).
- Read nPr when order matters, e.g. a ranking of winners.
- Read nCr when order doesn't matter, e.g. a team or a lottery.
Frequently asked questions
What is the difference between a permutation and a combination?
A permutation (nPr) counts arrangements where order matters. A combination (nCr) counts selections where it doesn't — e.g. lottery, teams.
How many combinations are there in a 6-from-49 lottery?
49C6 = 13,983,816. So the chance of getting all six numbers with one ticket is 1 in 13,983,816.
In how many ways can 5 books be arranged on a shelf?
5P5 = 5! = 5 × 4 × 3 × 2 × 1 = 120 ways. Enter n = 5 and r = 5 in the calculator.
What is a factorial?
n! is the product of all whole numbers from 1 to n. E.g. 5! = 120. By definition 0! = 1.