Combination: Definition, Formula & Examples

Combination: Definition, Formula & Examples
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  • 0:00 Combinations and Permutations
  • 1:11 Factorial Notation
  • 1:47 Combination Formulas
  • 2:52 Solving Combination Problems
  • 5:42 Lesson Summary
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Lesson Transcript
Instructor: Laura Pennington

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

In this lesson, we'll learn how to identify combinations of objects. We'll also look at various formulas that allow us to calculate the number of possible combinations in a given scenario.

Combinations and Permutations

In mathematics, combinations and permutations are normally studied at the same time because they are very similar. For instance, both permutations and combinations are collections of objects. But while a combination is a collection of the objects where the order doesn't matter, a permutation is an arrangement of a group of objects where the order does matter.

For example, suppose you were to choose four from a group of 20 people: Andrea, Alex, Sophie, and Nathan. If we said you chose Alex, Nathan, Sophie and Andrea, we'd still be talking about the same group of people. As the order in which we name them wouldn't matter, we'd be referring to a combination of four people.

Now, imagine that your employer just assigned you the following 4-digit employee identification code: 4793. If we changed the order of the digits to 9734, we'd end up with a different ID number. As the order of the digits does matter, 4793 is a permutation of four digits.

Factorial Notation

Before we get into the combination formulas, we first need to talk about n!, not in an excited fashion, which you may have assumed from the exclamation point, but in a mathematical context. In math, n!, pronounced n factorial, represents the product of all of the integers from n down to 1, as shown in the image below:

Permutation and Combination 1

For example, 4! = 4 * 3 * 2 * 1 = 24. We'll be using n! a lot when dealing with permutations and combinations.

Combination Formulas

When it comes to combination formulas, there are two scenarios we want to consider:

  1. Repetition is allowed
  2. Repetition is not allowed

To understand what these scenarios mean, let's revisit the group of people we discussed at the beginning of the lesson. Imagine that Alex was the first of the four we chose to be in our group. Once we've chosen Alex, we can't choose him again since he's already in the group. So, in this scenario, repetition is not allowed.

Now, pretend you're at a sushi bar that offers 11 different types of sushi, from which you can choose three. Suppose the first type of sushi you choose is salmon. As the menu allows you to have 3 pieces of sushi, your second and third choices could also be salmon. In this scenario, repetition is allowed.

It is important to understand whether or not repetition is allowed when determining which formula to use when solving combination problems. The formulas we use when dealing with combinations are described in the image below:


Solving Combination Problems

To solve problems involving combinations, we follow these steps:

  1. Make sure you are dealing with a combination problem, where order does not matter, and not a permutation problem
  2. Determine if repetition is allowed
  3. Use the appropriate formula based on what you found in the second step
  4. Substitute known numbers for the values in the formula, and perform the operations

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