# Permutation Calculator

Created by Bogna Szyk
Reviewed by Steven Wooding and Jack Bowater
Last updated: Dec 21, 2022

This permutation calculator is a tool that will help you determine the number of permutations in a set (often denoted as nPr). Read on to learn:

• The permutation definition;
• Permutation formula; and
• The relation between permutation and combination.

## Permutation definition

A permutation is the number of ways in which you can choose r elements out of a set containing n distinct objects, where the order of the elements is important.

For example, imagine that you have a deck of nine cards with digits from 1 to 9. You draw three random cards and line them up on the table, creating a three-digit number. How many distinct numbers can you create?

## Permutation formula

Luckily, you don't have to write down all of the possible numbers. You can use a formula that will allow you to calculate the number of permutations in no time:

$P(n,r) = \frac{n!}{(n-r)!}$

where:

• $P$ is the number of permutations;
• $n$ is the total number of elements in the set; and
• $r$ is the number of elements you choose from this set.

The exclamation mark represents a factorial. Check out our factorial calculator for more information on this topic.

You may also notice that, according to the permutation formula, the number of permutations for choosing one element is simply $n$. On the other hand, if you have to select all elements, the formula gets reduced to $P(n,n) = n!$.

Let's apply this equation to our problem with numbered cards. We need to find the number of ways to choose 3 out of 9 cards:

$P(9,3) = \frac{9!}{(9-3)!} = \frac{9!}{6!} = 504$

You can check the result with our nPr calculator.

## Permutation and combination

If you switch on the advanced mode, you will be able to find the number of combinations as well. Combinations are also ways to choose $r$ out of $n$ elements. Unlike in permutations, the order of the combinations doesn't matter.

For example, imagine a bag filled with twelve balls, each one of a different color. You pick five balls at random. How many distinct sets of balls can you get?

If you know the number of permutations, you can easily calculate the number of combinations, too:

$C(n,r) = \frac{P(n,r)}{P(r,r)} = \frac{P(n,r)}{r!}$

You can also check out dedicated tool: permutation and combination calculator.

## What next?

Make sure to try our probability calculator if you are interested in probability problems.

You can also try to solve a more difficult problem with our permutation calculator: if you have a deck of 9 cards with digits from 0 to 8, how many distinct valid numbers can you create by picking three cards at random? A number is not valid if it starts with zero.

Bogna Szyk
Set of objects
Total number of objects n
Sample size r
Number of selections
Permutations
Permutations with repetitions
Combination / Permutation generator
Show
Permutations
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