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GMAT Permutation & Combination - When to Add and Multiply

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If you want to gain admission to a top-ranked business school, you must take the GMAT. A global exam, GMAT assesses your ability to perform well in quant and reasoning. One integral part of the exam is permutation & combination and probability. These make up 10% of the examination. If you skip this blog, you may lose out on 10% of your marks!

Do you need clarification about whether to add or multiply the cases while answering GMAT permutation and combination questions? Do you often get incorrect answers because you added entities rather than multiplied them? If it’s yes to any of the above questions, read on to understand how to avoid confusion by using "AND" and "OR" on GMAT Permutation and Combination questions.

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GMAT Permutation and Combination - When to Add and Multiply

Concept 1: When to add – usage of the keyword “OR”

Students usually face problems with Permutation and Combination questions in the GMAT about whether to add up all cases or multiply cases. Let us understand this with two examples.

Sarah owns an ice cream shop. She offers various flavors of ice cream, pastry, and toppings. She has 4 different flavors of ice cream (vanilla, chocolate, strawberry, and mint), 5 different flavors of pastry (pineapple, chocolate, strawberry, mango, blueberry), and 3 different toppings (chocolate chips, sprinkles, and caramel sauce).

 Question: Customers can order an ice cream cone or a pastry and choose only one flavor. How many options does a customer have for selecting an ice cream or a pastry flavor? Solution: Since the customer can only choose one ice cream or pastry flavor, we use addition. Total possible cases of icecream = Vanilla OR Chocolate OR Strawberry OR Mint Number of ways Sarah can choose ice cream= 4 Total possible cases of pastry = Pineapple OR Chocolate OR Strawberry OR Mango OR Blueberry Number of ways Sarah can choose pastry = 5 Therefore, the total possible cases = 4 OR 5= 4+5 = 7

Takeaway:

 When dealing with situations involving two or more events, where one event doesn't affect the other or both events cannot occur together, then adding up all events is necessary.  To determine if an OR (Order of Events) is present, look for the word OR in the question and add the events accordingly.

Concept 2: When to Multiply – Usage of Keyword “AND”

 Question: Sarah offers ice cream cones with one flavor of ice cream and one topping. How many different combinations of ice cream cones with toppings can a customer order? Solution: Since the customer has to choose an ice cream flavor and a topping, multiplication will be used. Total possible cases = Ice Cream Flavor AND Topping Number of ice cream flavors = 4 Number of toppings = 3 Therefore, the total number of combinations of ice cream cones with topping = 4 AND 3= 4 (flavors) x 3 (toppings) = 12

Takeaway:

 When dealing with situations involving multiple events that can happen simultaneously, it is essential to multiply them.  To do this, look for the word "AND" in the question and if it is present, it indicates that you need to multiply the events.

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Concept 3: When the ‘AND’ or ‘OR’ keywords are not there

Let us understand the situations where the word 'AND' or 'OR' is not explicitly mentioned in a question with the help of three example questions from GMAT permutation and combination.

Example 1: Navigating A Maze

Consider a maze with several interconnected paths, like the one shown below. How many different ways are there to navigate from Start to Finish?

Solution:

 The keywords "AND" or "OR" are not explicitly mentioned here. So, let's analyze the situation step by step. Start to Junction A: There is only one direct path from Start to Junction A. Therefore, there is 1 way to reach Junction A from Start. Start => Junction A: 1-way Junction A to Junction B: From Junction A, we have two possible paths to reach Junction B: via Path X or Path Y. Since we can only choose one path, we use the word "OR". Number of ways from Junction A to Junction B = 2 Junction A => Junction B: 2 ways (via Path X OR Path Y) Junction B to Finish: Similarly, from Junction B, we have two possible paths to reach the Finish: via Path P or Path Q. Again, we use the word "OR". Number of ways from Junction B to Finish = 2 Junction B => Finish: 2 ways (via Path P OR Path Q) Now, to find the total number of ways to navigate from Start to Finish, we need to consider all the possible combinations: Total ways = Start to Junction A AND Junction A to Junction B AND Junction B to Finish = 1 way * 2 ways * 2 ways = 4 ways Therefore, there are 4 different ways to navigate from Start to Finish in the maze.

In this example, although the keywords "AND" or "OR" are not explicitly mentioned, we determine whether to add or multiply based on the events that occur simultaneously or not. Since each path choice at each junction is mutually exclusive, we multiply the options at each step to find the total number of ways to reach the Finish.

Example 2: Creating Passwords

Consider creating a 4-character password using the characters A, B, C, D, and E without repetition of characters. How many passwords can be made?

Solution:

 This question also does not explicitly mention the keywords "AND" or "OR". Let's analyze the situation: First Character: We have to fill in the first character of the password. Can we create a password with just the first character? No, we need to fill in all four characters. Therefore, we use the keyword "AND" and will multiply all the cases. Second Character: Since no repetition of characters is allowed, we have 4 remaining characters after filling the first slot. The second character can be chosen from these 4 remaining characters. Again, we use "AND" and multiply the cases. Third Character: After filling in the first two characters, we have 3 remaining characters. The third character can be chosen from these 3 remaining characters. We continue using "AND" and multiply the cases. Fourth Character: After filling in the first three characters, we have 2 remaining characters. The fourth character can be chosen from these 2 remaining characters. Once more, we use "AND" and multiply the cases. Now, let's consider each slot individually: First Character: Since any character can be chosen initially, there are 5 options. Second Character: After choosing the first character, there are 4 remaining options. Third Character: After choosing the first two characters, there are 3 remaining options. Fourth Character: After choosing the first three characters, there are 2 remaining options. To find the total number of possible passwords, we multiply the number of options at each slot: Total ways = 5 (options for the first character) * 4 (options for the second character) * 3 (options for the third character) * 2 (options for the fourth character) Total ways = 5 * 4 * 3 * 2 = 120 Therefore, 120 different 4-character passwords can be created using the characters A, B, C, D, and E without repetition.

Example 3: Forming a Task Force

Suppose a task force needs to be formed from a pool of 5 engineers and 4 designers. The task force will consist of 3 members. How many different task forces can be formed?

Solution:

 In this scenario, we can form the task force by selecting 3 engineers OR 3 designers OR a combination of 2 engineers and 1 designer OR a combination of 1 engineer and 2 designers. Let's break down the possibilities: 1. Selecting 3 Engineers: There are 5 engineers, and we need to select 3 of them. This can be done in 5C3 ways. 2. Selecting 3 Designers: Similarly, there are 4 designers and we need to select 3 of them. This can be done in 4C3 ways. 3. Selecting 2 Engineers and 1 Designer: For this combination, we need to choose 2 engineers from 5 engineers and 1 designer from 4 designers. This can be done in 5C2 x 4C1 ways. 4. Selecting 1 Engineer and 2 Designers: For this combination, we need to choose 1 engineer from 5 engineers and 2 designers from 4 designers. This can be done in 5C1 x 4C2 ways. Now, to find the total number of ways to form the task force, we add up all these possibilities: Total ways = 5C3 + 4C3 + (5C2 x 4C1) + (5C1 x 4C2) Let's compute each term: 5C3 = 5! /3!(5!-3!) = 10  4C3 = 4! /3!(4!-3!) = 4 5C2 = 5! /2!(5!-2!) = 10 4C1 = 4! /1!(4!-1!) = 4 5C1 = 5! /1!(5!-1!) = 5 4C2 = 4! /2!(4!-2!) = 6 Now, let's substitute these values into the equation: Total ways = 10+4+(10x4)+(5x6)  Total ways = 10+4+40+30 Total ways = 84 Therefore, there are 84 ways to form the task force from the pool of engineers and designers.

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Tips on How to solve GMAT Permutation and Combination Questions

Below are some tips that can be used while solving GMAT Permutation and Combination questions.

• Specific keywords can be an important factor in solving PnC questions. You can trust them, but you should use common sense to ensure you are on the right path.
• After reading the question, determine if the occurrences are dependent or independent. Making this inference will allow you to answer the question more efficiently.
• In cases when OR is involved, add up all of the events.
• If AND is involved, multiply all of the events.
• When the keyword is unavailable, attempt to jot down all the events first and then decide whether you need them together or separately.

Conclusion

GMAT Permutation and Combination seem confusing, but you can clear it with the right strategy and preparation. You can grasp the topic by starting with simple questions on Permutation and Combination and then upgrading your preparation. The above tips will help you understand ways to solve these questions easily.

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