About Combinations With Repetition Calculator
This calculator counts combinations with repetition, also called multisets: how many ways to choose r items from n kinds when the same kind can be chosen again and order does not matter. The answer is C(n + r - 1, r). It is free, needs no sign-up and runs in your browser, so nothing you enter is uploaded.
The classic picture is stars and bars: r identical stars placed into n boxes need n - 1 dividers, so you choose which r of the n + r - 1 positions are stars.
How to use Combinations With Repetition 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.
Combinations with repetition
Choose the r star positions among n + r - 1 positions.
C(n + r - 1, r) = (n + r - 1)! / (r! (n - 1)!)Worked examples
3 scoops from 5 flavours gives 35
C(7, 3) = 35.
10 identical sweets among 4 children gives 286
C(13, 10) = 286.
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.
When do I use repetition?
When the same kind of item can be taken more than once, such as scoops of ice cream or coins of one value.
Why n + r - 1?
It counts the r items plus the n - 1 dividers between the n kinds in a stars and bars diagram.
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.
- Combination CalculatorEnter n and r to count the ways to choose r items from n when order does not matter.
- Permutation CalculatorEnter n and r to count the ordered ways to choose r items from n.
- 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.