GMAT Permutation & Combination Practice Questions with Answers

The GMAT exam is a critical step for anyone pursuing a management education at top business schools. Among the various topics tested in the GMAT, Permutation and Combination are key concepts that can significantly impact your score. 

These concepts not only assess your mathematical ability but also your logical thinking and problem-solving skills. Let’s dig into GMAT Permutation and Combination Questions, offering you a comprehensive overview, useful formulas, and real GMAT-style practice questions with detailed solutions.

What is Permutation and Combination?

In the GMAT Quantitative section, Permutation and Combination are fundamental topics in probability and combinatorics. Understanding these two concepts is essential because they deal with how objects can be arranged or selected, which often appears in various forms in GMAT problems.

• Permutation:  It refers to the arrangement of objects in a specific order. For example, arranging the letters A, B, and C in different ways gives us permutations like ABC, ACB, BAC, etc.

• Combination: It refers to the selection of objects without considering the order. For example, selecting two letters from the set {A, B, C} gives the combination {A, B}, {A, C}, and {B, C}.

Both concepts play a critical role in understanding the structure of data, and mastering them is crucial for tackling GMAT questions involving arrangement and selection. Yocket Prep offers in-depth resources to help you master the Quantitative section of the GMAT, including Permutation and Combination.

GMAT Permutation and Combination Formulas 

To solve GMAT questions related to Permutation and Combination, it’s crucial to understand the following core formulas:

Permutation Formula: Pn,r=n!n−r!

Where:

• n = total number of objects
• r = number of objects to arrange
• ! denotes factorial (e.g., 5!=54321)

Example:
If you want to arrange 3 objects out of 5, you would calculate:

P5,3=5!5−3!=5!2!=5431=60

So, there are 60 ways to arrange 3 objects from 5.

Combination Formula: Cn,r=n!r!n−r!

Where:

• n = total number of objects
• r = number of objects to select
• ! denotes factorial

Example:
If you want to select 3 objects from a set of 5, you would calculate: C5,3=5!3!5−3!=543321=10
So, there are 10 ways to select 3 objects from 5.

GMAT Permutation and Combination Questions with Answers

Here’s a set of 15 GMAT-style questions on Permutation and Combination with detailed solutions:

Question 1: How many ways can 5 people be arranged in a row?

Solution:
This is a basic permutation problem. We are arranging all 5 people in a specific order. Using the permutation formula:

P5,5=5!5−5!=5!=54321=120

So, the number of ways to arrange 5 people in a row is 120.

Question 2: In how many ways can a committee of 3 people be selected from a group of 10 people?

Solution:
This is a combination problem because the order doesn’t matter. We use the combination formula:

C10,3=10!3!10−3!=1098321=120

So, there are 120 ways to select 3 people from 10.

Question 3: How many 4-letter words can be formed using the letters A, B, C, D, and E?

Solution:
This is a permutation problem where we are arranging 4 letters from a set of 5. Using the permutation formula:

P5,4=5!5−4!=5!=120

So, 120 distinct 4-letter words can be formed.

Question 4: How many different 3-digit numbers can be formed from the digits 1, 2, 3, 4, 5, and 6 without repetition?

Solution:
This is another permutation problem where we are selecting 3 digits from 6 and arranging them. Using the permutation formula:

P6,3=6!6−3!=654=120

So, there are 120 possible 3-digit numbers.

Question 5: How many ways can 4 men and 3 women be arranged in a row such that all the men sit together?

Solution:
Treat all 4 men as one unit. So, we have 5 units (the 4 men as one unit + 3 women). The number of ways to arrange these 5 units is:

P5,5=5!=120

Within the men’s unit, the 4 men can be arranged in 4!=24 ways.
Thus, the total number of arrangements is:

5!4!=12024=2880

So, there are 2880 ways to arrange them.

Question 6: How many ways can you select 2 red balls from 5 red balls and 3 blue balls?

Solution:
This is a combination problem because we are selecting 2 red balls from 5 red balls. Using the combination formula:

C5,2=5!2!5−2!=5421=10

So, there are 10 ways to select 2 red balls.

Question 7: How many ways can 4 different books be arranged on a shelf?

Solution:
This is a simple permutation problem where we are arranging 4 distinct objects (books) on a shelf. The number of ways to arrange 4 books is:

P4,4=4!=4321=24

So, the number of ways to arrange 4 different books is 24.

Question 8: How many ways can a 4-digit number be formed using the digits 1, 2, 3, 4, 5, 6 without repetition?

Solution:
This is a permutation problem where we need to arrange 4 digits out of 6. The number of ways to arrange the digits is:

P6,4=6!6−4!=6543=360

So, 360 different 4-digit numbers can be formed.

Question 9: In how many ways can 3 women be arranged in a line if they cannot sit next to each other?

Solution:
First, calculate the total number of ways to arrange all 3 women and 4 men without any restrictions:

P7,7=7!=5040

Now, calculate the number of ways in which all 3 women sit together. Treat the 3 women as a single unit, which gives us 5 units (4 men + 1 women’s unit). The number of ways to arrange these 5 units is:

P5,5=5!=120

Within the women’s unit, the 3 women can be arranged in 3!=6 ways.
Thus, the number of ways in which all 3 women sit together is:

1206=720

Finally, subtract this from the total number of unrestricted arrangements:

5040−720=4320

So, there are 4320 ways in which 3 women can be arranged such that they are not sitting next to each other.

Question 10: How many ways can you form a team of 2 men and 2 women from a group of 5 men and 6 women?

Solution:
This is a combination problem where we select 2 men and 2 women. We calculate the combinations separately:

• Number of ways to select 2 men from 5:

C5,2=5!2!5−2!=10

• Number of ways to select 2 women from 6:

C6,2=6!2!6−2!=15

Thus, the total number of ways to form the team is:

1015=150

So, there are 150 ways to form a team of 2 men and 2 women.

Question 11: How many ways can 3 vowels be selected from the 5 vowels A, E, I, O, U?

Solution:
This is a combination problem where we select 3 vowels from a set of 5 vowels. Using the combination formula:

C5,3=5!3!5−3!=543321=10

So, there are 10 ways to select 3 vowels.

Question 12: How many ways can 4 objects be arranged in a circle?

Solution:
When arranging objects in a circle, the number of arrangements is given by:

Pn,n=n−1!1=n−1!

For 4 objects:

P4,4=3!=6

So, there are 6 ways to arrange 4 objects in a circle.

Question 13: How many different ways can you select 2 books from 5 books and arrange them on a shelf?

Solution:
This is a combination followed by a permutation. First, we select 2 books from 5 using the combination formula:

C5,2=5!2!5−2!=10

Then, we arrange the 2 selected books:

2!=2

So, the total number of ways to select and arrange the books is:

102=20

Thus, there are 20 ways.

Question 14: How many ways can 5 cards be dealt from a deck of 52 cards?

Solution:
This is a combination problem where we select 5 cards from 52. Using the combination formula:

C52,5=52!5!52−5!=525150494854321=2598960

So, there are 2,598,960 ways to deal 5 cards from a deck of 52 cards.

Question 15: How many ways can you arrange the letters in the word "LEVEL"?

Solution:
This is a permutation problem with repeated objects. The number of distinct arrangements is given by:

n!k1!k2!

Where n is the total number of objects, and k1,k2, are the frequencies of repeated objects. For "LEVEL":

P5,5=5!2!2!=5432122=30

So, there are 30 distinct ways to arrange the letters in the word "LEVEL".

How to solve GMAT Permutation and Combination Questions? 

Here are the refined steps to approach GMAT Permutation and Combination questions effectively:

1.Understand the Problem Type

First, determine whether the problem involves arranging items (permutation) or selecting items without regard to the order (combination).

2. Recognise Key Elements:

• For Permutation: Focus on the total number of items (n) and how many items are being arranged (r).
• For Combination: Focus on the total number of items (n) and how many items are being chosen (r) without regard to the order.

3. Use the Correct Formula:

• Permutation Formula: Pn,r=n!n−r!
• Combination Formula: Cn,r=n!r!n−r!

4. Identify Special Conditions:

• Check for special conditions, such as repeated objects or circular arrangements, which may require adjustments in your formula:
• Repeated objects: Use the formula accounting for repetitions.
• Circular arrangements: Use the formula n−1! instead of n! for circular arrangements.

5. Calculate Factorials:
Factorials are a key component in both formulas, so practise calculating n! and n−r! quickly and accurately.

6. Simplify the Expression:
After applying the formula, simplify the factorial expressions to make the calculation easier and faster.

7. Verify Your Answer:
Double-check whether the problem requires ordered (permutations) or unordered selections (combinations) and confirm the final answer makes sense within the problem context.

From the Desk of Yocket

Understanding Permutation and Combination for the GMAT is crucial to boost your quantitative skills and improve your test score. These topics not only challenge your mathematical abilities but also test your logical reasoning and problem-solving approach. By focusing on understanding the core formulas and practising with real GMAT-style questions, you can gain confidence and accuracy in tackling these problems. Yocket Prep provides structured prep materials and expert guidance to help you understand these topics with ease.