Permutations


Suppose a club of six people wants to choose a council of three officers: a president, a vice president, and a treasury. How many ways are there to choose the council from the six people?

To answer this, lets’s visualize these three positions:

There are six different people who could be chosen as president:

Once a president is chosen, there are five people left who could be chosen as vice president:


And once a president and vice president have both been chosen, there are four people left who could be chosen as treasury:


To find the number of possible ways to choose the council, we simply multiply these numbers together:

6 * 5 * 4 = 120

There are 120 different ways that to make a three-person council from a 6-person group.

In statistical terms, we say there are 120 permutations

What is a Permutation?

A permutation is just an arrangement of objects where the order is important. 

The number of permutations of objects taken at a time, denoted as nPcan be found using the formula:

nPr  =  n!  / (n-r)!

For example, the number of ways three people can be chosen from a group of six is:

6P3 = 6! / (6-3)!  =  (6*5*4*3*2*1) / (3*2*1)  = 720 / 6 = 120

Let’s walk through a couple more examples of finding permutations.

Finding Permutations

Example 1: A group of 15 students are trying out for a musical that only has 4 roles . How many different ways are there to choose 4 roles from these 15 students?

Using the visual  approach:


There are 15 students who could land the first role. Then once that role has been filled, there are 14 remaining students who could land the second role. Then 13 students left to land the third role. Then 12 students left to land the fourth role. We simply multiply these numbers together to find the total number of ways to choose 4 roles.

Using the formula:

nPr  =  n!  / (n-r)!

In this case, n = 15 and r = 4. Plugging these into the formula we get:

15P4  =  15!  / (15-4)!  =  15! / 11!  = 1,307,674,368,000 / 39,916,800  = 32,760

Example 2: Jessica has seven plants in her garden. She must choose three of the plants, one to put in the front yard, one to put in the side yard, and one to put in the back yard. How many different ways are there for her to choose three plants from the seven plants?

Using the visual  approach:


There are seven plants that could be chosen for the front yard. Then once a plant has been chosen for the front yard, there are six remaining plants that could be chosen for the side yard. Once both the front yard and side yard plants have been chose, there are five remaining plants that could be chosen for the back yard. We simply multiply these numbers together to find the total number of ways to choose three plants.

Using the formula:

nPr  =  n!  / (n-r)!

In this case, n = 7 and r = 3. Plugging these into the formula we get:

7P3  =  7!  / (7-3)!  =  7! / 4!  = 5,040 / 24= 210

Helpful Tools

Combination and Permutation Calculator
Factorial Calculator

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