If you’re preparing for the GMAT to apply to MBA programs, you likely know the importance of having a strong Quant score. At the same time, the scoring of the GMAT Quant section is tough. So, to achieve your target score, you may have to master basically every Quant topic, including Combinatorics, which involves GMAT Combinations and Permutations questions.
At the same time, hearing about the formulas and strategies, it’s easy to get the impression that mastering Combinatorics will be challenging. However, the truth is that, by learning the underlying logic of how these questions work, we can readily master this topic. So, in this post, I’ll cover GMAT Combinations and Permutations basics to set you on your way to understanding the topic while keeping it simple.
Here's what we'll cover:
Let’s start by discussing the principle that is the foundation of Combinatorics.
The Fundamental Counting Principle
All of Combinatorics is based on the fundamental counting principle, or FCP. The FCP states that, when considering sets of choices together, we multiply the number of choices in the sets to find the total number of possible outcomes.
In other words, the FCP is that, if there are a ways to do one thing and b ways to do another, there are a × b ways to do both together.
Let’s consider a basic example that illustrates how the FCP works.
An Example of the FCP in Action
Let’s say that we have 2 choices of pants and 3 choices of tops, and we want to figure out how many outfits can be created that include a pair of pants and a top.
We can find the number of possible outfits by multiplying the number of pants choices by the number of tops choices.
Pants: 2 Choices
Tops: 3 Choices
2 × 3 = 6 Possible Outfits
We can confirm this and see how it works by writing out the combinations, using P1 and P2 as the 2 pants choices and T1, T2 and T3 as the 3 tops choices.
P1T1 P1T2 P1T3
P2T1 P2T2 P2T3
We see that, for each of the 2 choices of pants, there are 3 choices of tops. So, we end up with a total of 6 ways to choose pants and tops together.
This basic way of using the FCP can be applied to various situations with varying numbers of choices.
Let’s now use the FCP to answer an example question.
FCP Example Question
On the menu of a restaurant, there are 6 different appetizers, 10 different entrées, and 5 different desserts. How many different three-course meals can be created with those items?
(A) 21
(B) 30
(C) 60
(D) 160
(E) 300
Explanation
In this case, for each of the 6 appetizer choices there are 10 entrée choices. Then, for each of the 10 entrée choices there are 5 dessert choices.
So, the total number of possible meals is 6 × 10 × 5 = 300.
The correct answer is (E).
If you understand that question, you’re on your way to mastering GMAT Combinatorics.
KEY FACT:
The fundamental counting principle is that, if there are a ways to do one thing and b ways to do another, there are a × b ways to do both together.
OK, we’ve seen how a basic Combinatorics solution works. Let’s now discuss permutations.
The Basics of Permutations
To understand Permutations, let’s start with a basic example.
The previous examples involved combinations of items, and the order in which the items were chosen didn’t matter. All that mattered was which items were chosen.
For example, it didn’t matter whether a particular pair of pants was chosen before or after a particular top. Either way, the same outfit was created.
At the same time, in the cases of some Combinatorics questions, the order of the elements does matter. Combinatorics questions in which order matters are Permutations questions.
For our basic Permutations example, let’s press a license plate.
For this example, let’s say that the license plate has 4 characters in 4 slots. Let’s also say that each of the outer two slots can be filled with any of the letters A through Z, and each of the inner two slots can be filled with any of the digits 0 through 9.
Here’s an example of a plate that could be created in that manner:
A27B
To determine how many different plates can be created, we start with 4 slots for the 4 characters.
_____ _____ _____ _____
Now, Since there are 26 letters of the alphabet, there are 26 ways to fill each of the two outer slots.
_26 _ _____ _____ _26__
Similarly, since there are 10 digits, there are 10 ways to fill each of the two inner slots.
_26 _ _10__ _10__ _26__
So, there are 26 choices of letters to put in the first slot. For each of those, there are 10 choices of digits to put in the second slot. For each of those, there are 10 choices for the third slot. For each of those, there are 26 choices for the fourth slot.
So the number of different plates we can create is the following:
26 × 10 × 10 × 26 = 676 × 100 = 67,600
That’s a lot of different plates, but the math is pretty straightforward.
KEY FACT:
When counting permutations, we consider the order in which elements appear.
To lock in this concept, let’s do a practice question that works like the above example.
A Basic Permutations Practice Question
A company plans to create 5-character security codes, each of which will have the following format: The first, third, and fifth characters will be one of the digits from 0 to 9, and the second and fourth characters will be one of the letters from A to E. How many such codes can be created?
(A) 40
(B) 1,000
(C) 12,500
(D) 25,000
(E) 67,600
Explanation
Since the codes have 5 characters, we start with 5 slots.
_____ _____ _____ _____ _____
The first, third, and fifth characters are digits from 0 to 9. So, we have 10 choices for each of those slots.
_10__ _____ _10__ _____ _10__
The second and fourth characters are letters from A to E. So, we have 5 choices for each of those slots.
_10__ __5__ _10__ __5__ _10__
To find the total number of possible codes, we multiply the choices.
10 × 5 × 10 × 5 × 10 = 25,000
So, the correct answer is (D).
Now let’s take this to another level with a more advanced Permutations question along the lines of something we might see on the GMAT exam.
A More Advanced Permutations Example
We have 10 books and a shelf that can hold 3 of them. How many arrangements of books can we put on the shelf?
(A) 27
(B) 30
(C) 72
(D) 120
(E) 720
In this case, unlike in the previous example, we don’t have choices that we can use more than once in a given arrangement. Instead, we have 10 books, each of which we can only use one time each in a particular arrangement. In other words, once we put a book in the first slot, we can’t put that same book in the second or third slot.
At the same time, we are still working with arrangements in which the order of the books matters. In other words if we change the order in which the books are arranged in the three slots, we will create a new arrangement, or permutation.
You may associate the word “permutations” with complexity. However, the process we use for answering this question is actually pretty simple.
Explanation
We start by determining how many slots we have. Since the shelf can hold 3 books, we have 3 slots.
_____ _____ _____
Now, to determine how many permutations are possible, we just see how many choices of books we have for each slot and multiply the choices, basically using the FCP.
We have 3 slots and 10 books, and we could fill the slots in any order. Let’s keep this simple though. Let’s fill the slots from left to right.
To fill the first slot, we could choose any of the 10 books.
_10__ _____ _____
Now, once a book has been chosen to fill that first slot, there are only 9 books left. So, for each of the 10 choices for the first slot, we have 9 choices for the second slot.
_10__ __9__ _____
After the second slot has been filled, for each of those 9 choices, we are left with 8 books from which to choose to fill the third slot.
_10__ __9__ __8__
Now, to see how many arrangements are possible, we just multiply the number of choices we had for the 3 slots:
10 × 9 × 8 = 720
So, there are 720 possible arrangements that can be created through choosing 3 books from 10 and placing them in the 3 slots.
The correct answer is (E).
What we just did is the basis of answering most GMAT Permutations problems. So, we can see that this topic doesn’t have to be complicated.
Let’s now continue by taking a step to see how Permutations and Combinations can be related.
A Step Toward Understanding Combinations
OK, check this out. What if we had a scenario similar to the previous one except in that there were just 3 books to be arranged in the 3 slots on the shelf? We need to understand this scenario in order to see how to count combinations, where the order does not matter.
We start with 3 slots:
_____ _____ _____
Now, in this scenario,, there would be 3 books to choose from for the first slot. For each of those 3 choices, there would be 2 to choose from for the second slot and then just 1 left to pop into the third slot.
__3__ __2__ __1__
So, 3 books in 3 slots can be arranged in 3 × 2 × 1 = 6 ways.
OK, now, how is this related to combinations? Let’s see.
How to Go From Permutations to Combinations
Let’s consider something. What if the order did not matter and we had 3 books to put into a box that holds 3 books? We would just put the 3 books into the box and be done.
So, if order does not matter, then as far as we are concerned, there is just 1 way to put 3 books into a box that holds 3 books.
Similarly if we had a box that holds 10 books and we didn’t care about the order, we would just put the books in the box. 10 books, 1 box, 1 way.
Now, let’s again consider the 10 books we were choosing 3 from. In the previous example, we cared about the order. We were arranging the books on the shelf.
So, in the previous example, we were creating permutations of books. Now let’s just choose combinations of books.
To choose combinations of 3 books out of the 10 books, we can just choose 3 books and put them into a box without being concerned about order.
When I was first learning about this, I actually tried to figure out a formula myself, and figuring out the formula on my own was tough. But then, I looked it up and found out that calculating the number of possible combinations can be done pretty easily.
Here’s how it works.
First Calculate the Number of Permutations
To calculate the number of combinations of 3 books chosen from 10, we first calculate the number of permutations of 3 books chosen from 10. So, we do what we did when putting books on the shelf, arriving at 10 × 9 × 8 = 720 arrangements.
Now, let’s consider something. Those permutations include arrangements of combinations of 3 books. In other words, for every set of 3 books that we choose, there are multiple arrangements. All those arrangements add up to all the permutations.
The issue is that, this time, we are not concerned with permutations. We don’t care about all the arrangements of sets of 3 books. We are just putting books into a box, and the order doesn’t matter. So, each set of 3 books is just considered just 1 combination.
So how do we get from the number of permutations to the number of combinations?
Calculating the Number of Combinations From the Number of Permutations
To get from the number of permutations to the number of combinations, we just divide to eliminate the multiple arrangements of each combination. In other words, we divide to eliminate the duplicates.
Makes sense, right? But how do we know what number to divide by?
To get from the number of permutations, in which order matters, to the number of combinations, in which order does not matter, we need to divide by the number of arrangements of each combination of 3 books. So, what is that number?
From what we did earlier, in the step toward understanding combinations, we know that a set of 3 books can be arranged in 6 different ways. The number of permutations includes all of those ways. So, to convert our set of permutations to a set of combinations, we divide by 6.
720 ÷ 6 = 120
So from 10 books, combinations of 3 books can be chosen 120 ways.
PRO TIP:
One way to calculate the number of possible combinations of choices is to calculate the number of permutations and then divide to eliminate duplicates.
Let’s wrap up by considering an example question in which we can apply these concepts again.
A Combinations Example Question
There are 10 members of a board, and 4 must be selected for a special committee. How many different committees of 4 board members could be created?
(A) 30
(B) 40
(C) 210
(D) 720
(E) 5,040
Explanation
To answer this question, we can start by finding the number of possible permutations of 4 members out of 10.
We start with four slots.
_____ _____ _____ _____
There are 10 members who could be selected for the first slot.
_10__ _____ _____ _____
There are 9 members left who could be selected for the second slot.
_10__ __9__ _____ _____
There are then 8 members who could be selected for the third slot.
_10__ __9__ __8__ _____
Finally, there are 7 members who could be selected for the fourth slot.
_10__ __9__ __8__ __7__
So, the number of possible permutations of 4 members out of 10 is the following:
10 × 9 × 8 × 7 = 5,040
Now, to find the number of combinations, we need to divide out the duplicates.
Each combination of 4 members can be arranged in the following number of ways:
4 × 3 × 2 × 1 = 24
So, to find the number of possible committees, we divide the number of permutations by 24.
5,040/24 = 210
So, the correct answer is (C).
Let’s now review the key takeaways from what we’ve discussed.
GMAT Combinations and Permutations: Key Takeaways
In our discussion of Combinatorics, we’ve seen the following:
- The foundation of Combinatorics is the fundamental counting principle.
- To calculate the number of possible Combinations or Permutations of sets of choices, we multiply the number of choices in each set to find the total number of possible outcomes.
- In a question involving Permutations, the order of the elements chosen matters.
- In a question involving Combinations, the order of the element chosen does not matter.
- In many cases, we can calculate a number of possible Combinations by first calculating the number of possible Permutations and then dividing out the duplicates.
So, that’s pretty much it. Sure, to master GMAT Combinations and Permutations questions, you’ll need to know some formulas and other things in addition to what we’ve discussed here. All the same, understanding these basic principles, you’re well on your way to rocking Combinatorics on test day.
For additional insights you can use to master the GMAT, see this post on how to score 705+ on the GMAT.
