About Permutation and Combination Calculator
This permutation and combination calculator is free, needs no sign-up and runs in your browser, so nothing you enter is uploaded. Enter n, the number of items to choose from, and r, the number you choose. It gives the number of permutations (order matters) and combinations (order does not matter), written out with factorials.
It also handles the repetition cases: permutations with repetition, n to the power r, and combinations with repetition, C(n + r - 1, r). A separate mode counts the distinct arrangements of a word or a set with repeated items, such as MISSISSIPPI, using n! divided by the factorial of each repeat count.
Answers are exact. Results are computed with big-integer arithmetic, so 52 choose 5 is 2,598,960 and 100 choose 50 shows every digit rather than a rounded 1.0089e29. When n is 8 or less the actual arrangements are listed, so you can see what is being counted.
How to use Permutation and 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.
Permutation and combination formulas
A permutation counts ordered selections, so ABC and CBA are different. A combination counts unordered selections, so they are the same. Each combination of r items can be ordered in r! ways, which is why nCr is nPr divided by r!.
nPr = n! / (n - r)!
nCr = n! / (r! (n - r)!)
with repetition, ordered: n^r
with repetition, unordered: C(n + r - 1, r)
repeated items: n! / (n1! n2! ... nk!)Worked examples
10 choose 3 gives 120
10! / (3! x 7!) = (10 x 9 x 8) / (3 x 2 x 1) = 720 / 6 = 120.
5 permute 3 gives 60
5 x 4 x 3 = 60.
5-card hands from a 52-card deck gives 2,598,960
Order does not matter in a hand, so this is 52 choose 5.
A 3-digit code from the digits 0 to 9, repeats allowed gives 1,000
10^3 = 1,000.
3 scoops from 5 flavours, repeats allowed, order ignored gives 35
C(5 + 3 - 1, 3) = C(7, 3) = 35.
Arrangements of the letters in MISSISSIPPI gives 34,650
11! / (4! x 4! x 2! x 1!) = 39,916,800 / 1,152.
Good to know
Permutation or combination: ask whether order matters
If swapping two chosen items gives a different outcome, use permutations. Examples are a podium of first, second and third place, a password and a seating plan. If swapping them gives the same outcome, use combinations. Examples are a committee, a hand of cards and lottery numbers.
The word "combination lock" is the famous exception: a real lock cares about order, so it is a permutation.
Limits of the calculator
n and r are limited to 1000 so that the page stays fast. Without repetition, r must not be larger than n. The listing is shown only when n is 8 or less and shows at most 500 entries, labelled with letters rather than your own item names.
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 the difference between a permutation and a combination?
In a permutation the order matters, in a combination it does not. Choosing 3 people from 10 for a committee is a combination. Choosing a president, a secretary and a treasurer is a permutation.
How do I calculate nCr?
Use n! divided by r! times (n - r)!. For 10 choose 3 that is 10 x 9 x 8 / (3 x 2 x 1) = 120.
How do I calculate nPr?
Multiply r numbers counting down from n. For 5P3 that is 5 x 4 x 3 = 60, which equals 5! / 2!.
What does repetition allowed mean?
It means the same item can be chosen more than once, such as digits in a code. Ordered, that gives n to the power r. Unordered, it gives C(n + r - 1, r).
How many ways can I arrange the letters of a word with repeats?
Divide the factorial of the number of letters by the factorial of each repeat count. For MISSISSIPPI that is 11! / (4! x 4! x 2!) = 34,650.
Are the answers exact?
Yes. They use big-integer arithmetic, not floating point, so even very large answers show every digit.
Why is r not allowed to be larger than n?
Without repetition you cannot choose more different items than you have. Switch to a repetition mode if items can be reused.
Popular Permutation and Combination Calculator pages
- 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.
- 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.