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Permutation Calculator: How Many Ways Can You Order a Selection?

Written by:
PK

Below is a permutation calculator, which will calculate the number of permutations, or ordered sets you can choose from a larger whole. Enter the number of things in the set n and the number you need to choose in your sample r and we'll compute the number of permutations.

If you don't actually care the order of the selection, use the combination calculator (or change the input in the tool).

Permutation Calculator

What is a Permutation?

A permutation tells you how many ways there are to arrange – and usually also, to choose a subset of – a set.

If you already have an ordered set, the number of permutations tells you how many ways there are to arrange those members. If you are choosing a subset from a larger whole, it means how many ways you can choose the subset, and also how you can arrange your choice.

Permutations vs. Combinations

Permutations are closely related to the concept of combinations. Combinations, however, are order agnostic. That is – combinations refer to how many subsets you can choose out of a whole set without caring about the order.

The best way to remember is: "a combination lock is a lie". Since you can't enter the code into a lock in any order, a combination lock is actually a permutation lock. That is, if your combo is 4-26-3 you have to enter it in exactly that order to open the lock.

Formula for Permutations

The formula for the calculating permutations is:

\frac{n!}{(n-r)!}

Where:

  • n – the size of the set
  • r – the size of the subset you are choosing

Note that the factorial of 0 is 1. In the case where you are not picking a subset and only reordering the larger set, you only need to calculation n!.

Example Permutation Calculation

Let's say you have 4 letters: A, B, C, D

How many permutations are there when choosing a 2 letter subset?

\frac{4!}{(4-2)!}=\frac{24}{(2)!}=12

At 12 permutations, this one is small enough to spell out each one (and verify it by exhaustion). This is every possible order of 2 letters from A, B, C, D:

  • AB AC AD
  • BA BC BD
  • CA CB CD
  • DA DB DC

And there you go – 12 permutations.

Using the Permutation Calculator

To compute the total number of permutations, first enter "n", the total number of things in your set. Next, enter "r" which is how large of a subset you would like to calculate. (If you aren't taking a subset, r can be the same as n).

After that, leave the "Permutations or Combinations?" pulldown menu on Permutations, unless you don't care about order.

Hit Calculate and we'll let you know how many permutations there are.

Enjoyed that? Visit our other calculators and tools.

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