About Combination Calculator
This combination calculator finds nCr, also written C(n, r) or "n choose r": the number of ways to choose r items from n when the order does not matter. It shows the factorial formula with your numbers and an exact answer. It is free, needs no sign-up and runs in your browser, so nothing you enter is uploaded.
Combinations are symmetric: choosing r items to take is the same as choosing n - r to leave, so nCr equals nC(n - r). When n is 8 or less, the actual combinations are listed with letters.
How to use Combination Calculator
- Choose a mode: both nPr and nCr, one of them, a repetition case, or arrangements with repeats.
- Enter n and r as whole numbers. r cannot be larger than n unless repetition is allowed.
- Read the exact answer. For very long answers the copy button copies every digit.
- Open the working to see the formula with factorials and the cancelled product.
- For small n, scroll to the list to see the permutations or combinations written out with letters A, B, C.
Features
- nPr and nCr with exact BigInt results, up to n = 1000.
- Permutations and combinations with repetition allowed.
- Arrangements of a word or a multiset: n! / (n1! n2! ...).
- Factorial working shown in KaTeX, with the cancellation step.
- The actual arrangements listed when n is 8 or less, capped at 500 shown.
- Copy the full digits of a big answer. Works on a phone.
Combination formula
Divide n! by r! and (n - r)!.
nCr = n! / (r! (n - r)!)Worked examples
10 choose 3 gives 120
10 x 9 x 8 / (3 x 2 x 1) = 120.
52 choose 5 gives 2,598,960
The number of 5-card poker hands.
49 choose 6 gives 13,983,816
The number of 6-number lottery tickets from 49.
20 choose 10 gives 184,756
The middle of the 20th row of Pascal's triangle.
Privacy: how it works
- Everything is calculated in your browser tab. The numbers and words you enter are not sent to Ulkaa or to any other server.
Tips
- nCr equals nC(n - r), so 100 choose 98 is the same as 100 choose 2.
- 0! is 1, which is why choosing none or all of the items gives exactly one way.
- For "at least" questions, count the opposite case and subtract from the total.
- For repeated letters, type the word. The calculator counts each letter for you.
Frequently asked questions
Is it really free?
Yes. There is no price, no trial and no credit card. Ulkaa Tool does the work with your own device, so there is no server bill to pass on to you.
Do I need an account?
No. There is no sign-up and no login. Open the page and use it. An Ulkaa account is not needed for anything on this tool.
Does it run in my browser?
Yes. Your numbers are processed by code running in your browser tab, on your own device.
Is my data uploaded?
No. Your numbers are never sent to Ulkaa or to anyone else. You can check this yourself by watching the Network tab in your browser developer tools while you use the tool.
What is nCr?
It is the number of ways to choose r items from n when order does not matter, also called a binomial coefficient.
What is nC0 and nCn?
Both are 1: there is one way to choose nothing and one way to choose everything.
More Permutation and Combination Calculator pages
- Permutation and Combination CalculatorEnter n and r to get nPr and nCr with exact answers, the factorial working and the arrangements listed when they are few.
- Permutation CalculatorEnter n and r to count the ordered ways to choose r items from n.
- Combinations With Repetition CalculatorCount unordered selections of r items from n kinds when an item can be chosen more than once.
- Permutations With Repetition CalculatorCount ordered selections of r items from n kinds when items can repeat.
- Permutations of a Word CalculatorType a word to count how many different ways its letters can be arranged.