Permutations and Combinations Calculator
Enter n (total items) and r (items chosen) to calculate permutations (order matters) and combinations (order doesn't matter).
Last updated: September 5, 2026
Frequently asked questions
What's the difference between a permutation and a combination?
Permutations count arrangements where order matters (like a race's 1st/2nd/3rd place); combinations count selections where order doesn't matter (like picking a 3-person committee). The same group of items counted as a permutation is always larger, since it counts each ordering separately.
What is 10 choose 3 (10C3)?
120 - using nCr = n! / (r! × (n-r)!): 10! / (3! × 7!) = 3,628,800 / (6 × 5,040) = 120.
What is 10P3?
720 - using nPr = n! / (n-r)!: 10! / 7! = 10 × 9 × 8 = 720.
Why is nPr always nCr multiplied by r!?
Every combination of r items can be arranged in r! different orders, and each of those orderings counts as a separate permutation - so permutations = combinations × r!, or equivalently combinations = permutations ÷ r!.
What is 5 choose 5 (5C5), and why does it equal 1?
1 - there's only one way to choose all 5 items from a group of 5 (you simply take everyone), which the formula confirms: 5!/(5!×0!) = 1, since 0! is defined as 1.
How many different 4-letter arrangements can be made from 6 distinct letters?
360 - using nPr = n!/(n-r)! = 6!/(6-4)! = 720/2 = 360, since order matters for an arrangement (permutation).
Why is nCr always the same value whether you compute it as 'n choose r' or 'n choose (n-r)'?
Choosing r items to include is mathematically identical to choosing the (n-r) items to leave out - for example, 10C3 = 10C7 = 120, because both describe the same act of splitting 10 items into a group of 3 and a group of 7.