Permutation & Combination Calculator

Counts the ways to choose or arrange r items out of n: permutations (nPr), combinations (nCr), both with repetition, the factorial n! and circular arrangements. Use it to check probability and counting problems.

How to use

1. Enter the number of items in n and the number chosen in r. Whole numbers from 0 to 10,000 are accepted.
2. "Permutations", "Combinations", the versions with repetition, "Factorial" and the circular arrangements appear with one line of working each.
3. Under "List the outcomes", choose "What to list" to see every outcome, optionally with your own "Item names".
4. For words with repeated letters such as BANANA, enter the word or a list of counts under "Permutations with identical items".

How it works

• nPr = n! ÷ (n − r)!, nCr = n! ÷ (r! × (n − r)!), permutations with repetition = n^r, combinations with repetition = (n + r − 1)Cr, circular arrangements = (n − 1)!.
• Values are exact. Results with 16 or more digits are shown in scientific notation; "Show all digits" lets you copy every digit.
• Lists appear only for 100 outcomes or fewer. When r is larger than n, permutations and combinations are 0.

Questions and answers

What is the difference between a permutation and a combination?

A permutation counts order and a combination does not. Choosing a president and a vice president from 5 people gives 5P2 = 20; choosing any 2 representatives gives 5C2 = 10.

How many ways are there to pick 6 numbers out of 49?

Enter n = 49 and r = 6. "Combinations" shows 13,983,816, because the order of the numbers does not matter.

Why is 0! equal to 1?

There is exactly one way to arrange nothing, so 0! is defined as 1. This keeps formulas such as nCn = 1 correct in every case.

When do I use combinations with repetition?

When the same item can be chosen more than once and order does not matter. Choosing 2 scoops from 3 flavors gives (3 + 2 − 1)C2 = 4C2 = 6.