How to Solve 5 Choose 2

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  • 0:03 Steps to Solving the Problem
  • 2:45 Solution
  • 2:57 Application
  • 4:23 Lesson Summary
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Lesson Transcript
Instructor: Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

In this lesson, we'll learn how to find 5 choose 2. We'll look at two different ways to solve this and the steps involved. We'll end with a real-world example where we could use this process.

Steps to Solving the Problem

We want to know how to find 5 choose 2. This problem is often written using the notation 5 C 2 , and it means we're looking for how many ways there are to choose 2 objects from 5 objects when the order of the objects doesn't matter. This is called a combination of 2 objects chosen from 5 objects. In general, a combination is a group of objects where order does not matter. For instance, if I have a soup made of chicken, celery, and broth, it is the same soup if I say it's made of celery, chicken, and broth. The order of the ingredients does not matter.

First, let's look at solving this problem using a formula. If we want to find n choose r, we use the combination formula, as follows.

5 C 2 1

You may be wondering what those exclamation points mean. In mathematics, ! represents a factorial. If we want to find n!, we multiply all the integers from n counting backward down to 1. This is shown in the factorial formula.

5 C 2 2

To use the formula to solve the problem, we first identify n and r, and then plug those values into our formula. In our problem, we want to find 5 choose 2. Therefore, n = 5 and r = 2, so we plug those values into our formula and simplify the formula as shown.

5 C 2 3

Another way to solve this problem is to actually list the possible combinations of 2 objects from 5 objects and count them. This process wouldn't work well if the numbers were larger, but in this case, we can get away with it. To solve this way, consider 5 objects A, B, C, D, and E. Then we just list the possible pairs taking note that order doesn't matter, so, for instance, AB is the same as BA. Keeping this in mind, the possible combinations of 2 objects from the 5 objects are as follows.



In both of our solving processes, we see that 5 C 2 = 10. In other words, there are 10 possible combinations of 2 objects chosen from 5 objects.

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