Combinations


What is a Combination?

A combination is an arrangement of objects where the order is not important. 

For example, suppose we have a group of three friends: Zach, Ty, and AJ. We want to know all the possible combinations of two people we can form from these three friends.

In this example ,the combination of “Zach and AJ” is the exact same as “AJ and Zach.” These two groups are the exact same because they contain the same people. The order that we list the friends does not matter.

In total, there are three distinct groups we could form from these three friends:

1. Zach and Ty
2. Zach and AJ
3. Ty and AJ

The Combination Formula

The formula to find the number of combinations of objects taken at a time, denoted as nCr ,can be found by:

nC = n! / r!(n-r)!

In the previous example, n = 3 total friends, r = 2 friends per group, so the total number of combinations would be:

3C = 3! / 2!(3-2)!  =  (3*2*1) / (2*1)(1)  =  6 / 2  =  3

Let’s walk through some more examples of finding combinations.

Examples of Finding Combinations

Example 1: You won a ticket to a concert and you’re allowed to bring two friends along with you. Unfortunately,  you have five friends who want to come along. How many different groups of friends could you take with you?

In this case, there are five total friends (n = 5) and you want to select a group of two (r = 2).  Using the formula:

Total combinations = 5C  =  5! / 2!(5-2)!  =  5! / 2!(3!)  =  (5*4*3*2*1) / (2*1) * (3*2*1)  =  120 / 12  =  10

There are 10 different groups of friends you could take with you to the concert. 

Example 2: You are taking a trip to Alaska. You have four different pairs of shoes you want to bring, but only enough room in your suitcase for three shoes. How many different groups of shoes could you take with you?

In this case, there are four total pairs of shoes (n = 4) and you want to select a group of three (r = 3). Using the formula:

Total combinations = 4C  =  4! / 3!(4-3)!  =  4! / 3!(1!)  =  (4*3*2*1) / (3*2*1) * (1)  =  24 / 6  =  4

There are 4 different groups of shoes you could take with you.

Example 3: You are a basketball coach and you have six people trying out for your team that has five positions available. How many different groups of players could you select?

In this case, there are six total players  (n = 6) and you want to select a group of five (r = 5). Using the formula:

Total combinations = 6C  =  6! / 5!(6-5)!  =  6! / 5!(1!)  =  (6*5*4*3*2*1) / (5*4*3*2*1) * (1)  =  720 / 120  =  6

There are 6 different groups of players you could select.

Example 4: Five people want to get in a boat to cross a river. Unfortunately, the boat is only large enough to carry three people. How many different groups of people could get in the boat?

In this case, there are five total people (n = 5) and we want to select a group of three (r = 3). Using the formula:

Total combinations = 5C  =  5! / 3!(5-3)!  =  5! / 3!(2!)  =  (5*4*3*2*1) / (3*2*1) * (2*1)  =  120 / 12  =  10

There are 10 different groups of people that could get in the boat.

Helpful Tools

Combination and Permutation Calculator
Factorial Calculator

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