Permutation and Combination Calculator
Count the ways to arrange or choose r items from n: permutations (nPr) when order matters and combinations (nCr) when it doesn’t, with or without repetition, exactly, even for very large numbers.
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How do I calculate permutations and combinations?
Use permutations when order matters and combinations when it doesn’t. Choosing 3 items from 10 in order gives nPr = 10 × 9 × 8 = 720 ways; as unordered groups it gives nCr = 720 ÷ 3! = 120. With repetition allowed, ordered choices are nʳ and unordered ones C(n + r − 1, r).
- Enter n, the number of items to choose from, and r, how many you choose.
- Tick “Repetition allowed” if an item can be chosen more than once, as with digits in a PIN.
- Read the number of permutations (nPr, order matters) and combinations (nCr, order doesn’t).
Formulas & Methods
Permutations, nPr
Example: 10P3 = 10 × 9 × 8 = 720
Combinations, nCr
Example: 10C3 = 720 ÷ 6 = 120
With repetition
Example: n = 10, r = 3: 1,000 and 220
Key features
Both answers at once
Permutations and combinations side by side, so you can see how much order matters.
With or without repetition
Covers PIN codes and dice as well as committees and lottery draws.
Which one do I need?
Ask whether swapping two chosen items gives a different result. A race podium is a permutation: gold, silver and bronze for the same three runners in a different order is a different outcome. A lottery ticket is a combination: the same six numbers in any order are the same ticket, which is why there are 13,983,816 ways to pick 6 numbers from 49. A 4-digit PIN allows repetition: 10⁴ = 10,000 codes.
Frequently Asked Questions
Why is nCr always smaller than or equal to nPr?
Each group of r items can be put in order in r! ways, and nPr counts all of them separately. So nCr = nPr ÷ r!, which is smaller whenever r is 2 or more.
How many poker hands are there?
2,598,960: the number of ways to choose 5 cards from 52 when order doesn’t matter, 52C5.
What is 0! and why is nC0 equal to 1?
0! is defined as 1, and there is exactly one way to choose nothing: the empty selection. So nC0 = nP0 = 1 for any n.
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