Last Updated on April 26, 2023
After working with GRE students for many years, I have consistently seen that students struggle to recognize when to use permutations or combinations in any given combinatorics problem. In this article, we will discuss the differences between combinations and permutations, and practice some examples of each.
- What Is the Difference Between a Combination and a Permutation?
- When to Use Combination vs. Permutation
What Is the Difference Between a Combination and a Permutation?
Permutations and combinations are both methods for counting the number of ways we can select from collections of objects. The key difference between the two is that in permutations, the order matters, whereas in combinations, the order does not matter.
To explore this idea further, let’s consider a common scenario: selecting the members of a committee. When you’re choosing a group of people to form a committee, does the order of the selection matter? The answer may depend on what roles each of the committee members will have.
For example, suppose that Tabitha, Darlene, and Carlos are selected to form a committee of three people with equal responsibilities. In this case, the order of that selection does not matter. After all, whether the committee is made up of {Tabitha, Darlene, Carlos} or {Carlos, Tabitha, Darlene}, the committee is the same, right? In this case, because we see that order does not matter, we have a combination.
KEY FACT:
When we are choosing items from a set, if the order of selection does not matter, we have a combination.
Now suppose that we are selecting a committee that is made up of a president, vice president, and secretary, with the same three people as options. Once again, we are selecting three people for a committee. However, in this case, if we have Tabitha as president, Darlene as vice president, and Carlos as secretary, we see that this committee differs from one in which we have Darlene as president, Carlos as vice president, and Tabitha as secretary. Since the order matters in this scenario, we have a permutation.
KEY FACT:
When we are choosing items from a set, if the order of selection does matter, we have a permutation.
When to Use Combination vs. Permutation
Let’s now practice with a few examples that require us to differentiate between combinations and permutations. Remember, when analyzing these problems, we always want to ask ourselves, “Does order matter?” The answer to that question will allow us to decide whether we have a combination or a permutation.
Example 1
There are 15 books on a bookshelf. If 3 of the books are taken from the bookshelf to be moved to another shelf, in how many ways can they be selected?
Solution:
Let’s say that the 3 books selected are a biography, a poetry book, and a novel. The order in which those books are selected does not matter; we just need to know which 3 books we are moving. For example, if the order is {poetry-biography-novel} or {novel-poetry-biography} the same 3 books will move to the other shelf. Thus, we have a combination.
Example 2
Two students from a class of 20 are to be selected to be on the lunch committee. How many different groups of 2 students can be chosen?
Solution:
Let’s say that of the 20 students, Pablo and Myra are chosen. Does anything change if the order in which those students are selected is {Pablo-Myra} or {Myra-Pablo}? No; in either arrangement, the same 2 people will be on the lunch committee. The order does not matter, and so we have a combination.
Example 3
From a team of 30 soccer players, 4 players are to be selected and arranged in a line for a team photo. In how many ways can those 4 players be arranged?
Solution:
Suppose that Howie, Hunter, Olivia, and Gigi are the 4 selected players who will be arranged in a line for the photograph. We can see that an arrangement of {Hunter-Howie-Olivia-Gigi} differs from an arrangement of {Olivia-Hunter-Gigi-Howie}. (After all, those photos would look different!) So, order does matter, meaning that we have a permutation.
Example 4
At a bus station, there are 12 people waiting to board a bus. If 2 people from the 12 are selected to fill a window seat and an aisle seat, how many different seating arrangements are possible?
Solution:
Let’s say that Pauline and Marcos are selected to be seated. Putting Pauline in the window seat and Marcos in the aisle seat differs from putting Marcos in the window seat and Pauline in the aisle seat. Thus, the order does matter, and so we have a permutation.
After completing these four examples, you should have a clearer idea of the difference between a combination and a permutation. If you can harness this foundational skill, you will have a much easier time attacking combinatorics questions. If you would like to learn more about combinations and permutations, including how to solve all the types of these problems that you will encounter on the GRE, try a full-access trial of the Target Test Prep GRE Quant Course.
