FreeToolStation

Permutation Calculator and Combination Calculator

Count arrangements and selections, with the working.

Statistics Calculators Free, no sign-up Runs in your browser
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Permutations (order matters)
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Combinations (order does not)
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n factorial
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nPr / nCr
Ask whether swapping two of your chosen items gives a different outcome. If it does, order matters and you want the permutation. If it is the same outcome, you want the combination.

About the Permutation and Combination Calculator

Permutation and Combination Calculator works out how many ways there are to arrange or choose things, and shows the factorials that produced the answer.

The difference between the two is one question: does the order matter? A permutation counts arrangements, where first and second are different outcomes. A combination counts selections, where picking Ali then Sara is the same result as picking Sara then Ali. A race podium is a permutation. A committee is a combination. Getting this backwards is the single most common mistake in the topic, and it is why the page asks the question in those words before giving a number.

The formulas are nPr = n! / (n-r)! and nCr = n! / (r! x (n-r)!). The combination formula is the permutation divided by r!, which is exactly the number of ways to reorder the r things you chose, and that division is what removes the orderings you did not want counted.

Repetition is handled too, as a separate mode: arrangements with repetition allowed is n^r, and selections with repetition is (n+r-1)C r. Those come up in password strength, dice problems and combination-lock questions, and they give much larger numbers than the no-repetition versions.

Everything is worked out inside your browser. Nothing you enter is uploaded, logged or stored, which matters for the financial calculators in particular: the figures you put into them are nobody else's business.

Looking for something this does not cover? FreeToolsHub.org is another hub of free online tools that need no account, and it is worth a look if permutation and combination calculator is only part of what you are working on.

How to use the Permutation and Combination Calculator

  1. Say whether order mattersIf it does you want a permutation; if not, a combination.
  2. Enter n and rn is how many you are choosing from, r how many you are choosing.
  3. Choose whether repetition is allowedAllowed means the same item can be picked more than once.
  4. Read the answer and the workingThe factorials and the cancellation are shown.

Features

  • nPr and nCr, both with and without repetition
  • Full working with the factorials written out
  • Factorials of large n handled without losing precision
  • A plain-language test for choosing permutation or combination
  • Pascal's triangle relationship shown for combinations
  • Results given in full and in scientific notation when huge
  • Runs in your browser

Who uses this tool

The permutation and combination calculator is used by writers, developers, students, marketers and anyone else who needs the job done once without installing software. Common cases include:

  • Probability and statistics coursework
  • Counting possible lottery or draw outcomes
  • Working out how many passwords a character set allows
  • Planning seating or team arrangements
  • Checking a combinatorics answer

Frequently asked questions

How do I know which one I need?

Ask whether swapping two of your chosen items gives a different outcome. If it does, order matters and it is a permutation. If it is the same outcome, it is a combination.

Why is nCr smaller than nPr?

Because nCr groups together all the orderings of the same selection. It is nPr divided by r!, the number of ways those r items can be rearranged.

What is 0! and why is it 1?

There is exactly one way to arrange nothing: the empty arrangement. Defining 0! as 1 keeps the formulas working at the edges rather than breaking at r = n.

Will very large numbers be accurate?

The calculator cancels factorials before multiplying rather than computing them in full, so values stay exact far longer than a naive approach allows. Beyond the point where exact arithmetic is possible it says so instead of quietly rounding.

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Checked and rebuilt on 2026-10-14. The method this calculator uses is described in the section above, and corrections go through the contact page.