Problem 53 - Algebra for JEE ADVANCED

Question

A person always prefers to eat 'parantha' and 'vegetable dish' in his meal. How many ways can he make his platter in a marriage party if there are three types of paranthas, four types of 'vegetable dish', three types of 'salads' and two types of 'sauces"? a. 3360 b. 4096 c. 3000 d. none of these

Step-by-Step Solution

Answer

72 combinations; choice (d) none of these.

Step 1: Determine Choices for Paranthas

The person can choose from three types of paranthas. So, there are 3 choices for paranthas.

Step 2: Determine Choices for Vegetable Dishes

He can choose from four types of vegetable dishes. Hence, there are 4 choices for vegetable dishes.

Step 3: Consider Choices for Salads

There are three types of salads available. So, he has 3 choices for salad.

Step 4: Evaluate Choices for Sauces

There are two types of sauces. This gives him 2 choices for sauces.

Step 5: Calculate Total Combinations

To find the total number of ways he can fill his platter, multiply the number of choices for each category: \(3 \text{ (paranthas)} \times 4 \text{ (vegetable dishes)} \times 3 \text{ (salads)} \times 2 \text{ (sauces)} = 72\).

Step 6: Check Answer Against Options

The calculated total number of combinations is 72. Given options do not include 72, indicating the correct answer is (d) none of these.

Key Concepts

Permutations

Permutations are a fundamental concept in combinatorics. They involve rearranging or ordering elements from a set. For example, if you have three types of paranthas, arranging them in different orders or selecting one would be dealing with permutations. The number of permutations depends on whether the order matters. If it does, then every possible sequence is considered different.
To calculate permutations, you often use the factorial function, denoted by an exclamation mark (!) which multiplies a series of descending natural numbers. In simpler terms, if you have a set of 'n' items and choose to permute them all, the number of permutations is 'n!'.
However, in this exercise, we are not reordering a selection of paranthas, but choosing one. Thus, permutations are not the primary method needed here.

Combinations

Combinations are an essential part of combinatorics, but they differ from permutations. Unlike permutations, combinations focus on selecting items from a larger set without considering the order of selection. In the context of the exercise, choosing one type out of several options for each food item (such as three paranthas, four vegetable dishes, etc.) involves combinations.
The formula for combinations is given by \({n \choose r} = \frac{n!}{r!(n-r)!}\), where 'n' is the total number of items to choose from, and 'r' is the number of items to choose. But in simpler selections, like our example, it simplifies to just picking one from 'n' options directly.
This makes combinations a straightforward choice when the order does not matter, which applies to our dish selection scenario.

Multiplicative Principle

The multiplicative principle is a cornerstone of counting methods. It helps in determining the number of outcomes when dealing with multiple independent choices. When something needs to be chosen from various categories independently, you multiply the number of ways to choose each item. In the original exercise, this principle is utilized to calculate how this person can assemble his meal plate.
The choices available are: 3 types of paranthas, 4 types of vegetable dishes, 3 types of salads, and 2 types of sauces. Each choice is independent of the others. Therefore, by applying the multiplicative principle, we multiply all these choices together: \[3 \times 4 \times 3 \times 2 = 72\] This tells us there are 72 different ways to put together a meal. This comprehensive principle makes it easy to calculate the total number of combinations when there are several layers of decisions to consider.