GMAT Problem Solving- At a Sandwich Shop, There are 2 Breads, 3 Kinds of Meat, 4 Types of Cheeses

Question: At a sandwich shop, there are 2 breads, 3 kinds of meat, 4 types of cheeses, and 6 different sauces to choose from. Customers can choose only one of each. Assuming a customer has already chosen one kind of bread and one kind of meat for a sandwich, how many different sandwiches can be made from the other ingredients?

“At a sandwich shop, there are 2 breads, 3 kinds of meat, 4 types of cheeses,”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Quantitative Review”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

Solution and Explanation:

Approach Solution 1:
Given to us that, at a sandwich shop, there are 2 breads, 3 kinds of meat, 4 types of cheeses, and 6 different sauces to choose from. Customers can choose only one of each. We have to assume that a customer has already chosen one kind of bread and one kind of meat for a sandwich. It has asked how many different sandwiches can be made from other ingredients.
Therefore the question says out of the given four items -

2 breads
3 kinds of meat
4 types of cheeses
6 types of sauce

We can only choose one item from each kind items
This is a question of permutation and combination. The general formulas are given below:
No of permutations of n objects taking r at a time =\(^nP_r\) = \(\frac{n!}{(n-r)!}\)
No of combinations of n objects taking r at a time = \(^nC_r = \frac{n!}{r!(n-r)!}\)
But it is already given that customer has chosen one kind of bread and one kind of meat,
Therefore,
No. of ways to choose one kind of bread = 1 (because bread already chosen)
No of ways to choose the kind of meat = 1 (because bread already chosen)
No of ways of choosing 1 kind of cheese from four varieties = \(^4C_1\) = 4!/(4-1)!*1! = 4
No of ways of choosing 1 kind of sauce from six varieties of sauces = \(^6C_1\) = 6!/(6-1)!*1! = 6
Total number of sandwiches that can be made from the above ingredients = 1 * 1 * 4 * 6 = 24

Correct Answer: B

Approach Solution 2:

Given that there are 2 kinds of bread, 3 types of meat, 4 varieties of cheese, and 6 sauces available at a sandwich restaurant. Customers are limited to choose one of each. For the purpose of making a sandwich, we must assume that the diner has already decided on one type of bread and one type of meat. How many distinct kinds of sandwiches may be created using additional ingredients?
As a result, the query reads: "Out of the following four items:

2 breads
3 types of meat
4 varieties of cheese
6 different sauces

From each category, we can only select one thing.
Permutation and combination are at issue here. Below are the general formulas:
Number of n item permutations taking r at a time = \(^nP_r\) = \(\frac{n!}{(n-r)!}\)

No of combinations of n objects taking r at a time = \(^nC_r = \frac{n!}{r!(n-r)!}\)
Given that the customer has previously selected one type of bread and one type of meat, the number of combinations of n objects taking r at a time is equal to 1. (because bread already chosen)
There is just one way to pick the type of meat (because bread already chosen)
The number of ways to choose one type of cheese out of four options is 4!/(4-1)!
*1! = 4
The number of methods to choose one sauce type out of six sauce variants is 6!/(6-1)!
*1! = 6
The components listed above may be used to make a total of 24 sandwiches: 1 * 1 * 4 * 6

Correct Answer: B

Approach Solution 3:

Considering that a sandwich shop offers two types of bread, three choices of meat, four kinds of cheese, and six sauces. Customers can only select one of each. We must presume that the diner has already chosen on one type of bread and one type of meat in order to make a sandwich. How many different types of sandwiches may be made using extra ingredients?
The outcome is that the question is as follows: "Of the following four items:

2 breads
3 kind of meat
3 kind of meat types of cheese
6 distinct sauces

There is just one item we may choose from each category.
The key words here are combination and permutation. The general formulae are listed below:

number of r-at-a-time permutations of n items = \(^nP_r\)= \(\frac{n!}{(n-r)!}\)

number of ways to combine n items while removing r at a time = \(^nC_r = \frac{n!}{r!(n-r)!}\)
The number of combinations of n objects taking r at a time is equal to 1, given that the client has already chosen one type of meat and one type of bread. (because the bread is already selected)
There is just one method for choosing the sort of meat (because bread already chosen)
There are 4!/(4-1) methods to select one sort of cheese from the available choices.
*1! = 4
There are six different sauce types, hence there are six different ways to pick one of them: 6!/(6-1)!
*1! = 6
There are a total of 24 sandwiches that may be made with the ingredients mentioned above: 1 * 1 * 4 * 6

Correct Answer: B

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GMAT Problem Solving- At a Sandwich Shop, There are 2 Breads, 3 Kinds of Meat, 4 Types of Cheeses

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