Question

A person always prefers to eat paratha and vegetable dishes in his meal. How many ways can he make a platter in a marriage party if there are three types of paratha, four types of vegetables, three types of salads, and two types of sauces?

Answer 1

Hint:
Here, we have to find the number of ways in which he can make the platter. We will find the number of ways he selects parathas, the number of ways he selects vegetables, the number of ways he selects salads, and the number of ways he selects sauces. Then, we will multiply these to get the required answer.

Complete step by step solution:
The person prefers to eat paratha and vegetables.
This means that he will add at least one paratha and vegetable dish to his platter.
We will find the number of ways in which he can select a paratha, a vegetable dish, a salad, and a sauce, and then multiply them together to get the required answer.
First, let us find the number of ways he can add the parathas.
There are three types of parathas.
For each of the three parathas, he has two choices – whether to add it, or not.
This means that the number of ways in which he can select parathas is \[2 \times 2 \times 2 = 8\].
However, we know that he will add at least one paratha to his platter.
It is not possible that he chooses not to add any of the three parathas.
Therefore, we will subtract 1 way from the 8 possible ways to get the number of ways in which he will select the parathas, considering he will take at least one.
Thus, we get the number of ways as \[8 - 1 = 7\].
Next, let us find the number of ways he can add the vegetables.
There are four types of vegetables.
For each of the four vegetables, he has two choices – whether to add it, or not.
This means that the number of ways in which he can select vegetables is \[2 \times 2 \times 2 \times 2 = 16\].
However, we know that he will add at least one vegetable to his platter.
It is not possible that he chooses not to add any of the four vegetables.
Therefore, we will subtract 1 way from the 16 possible ways to get the number of ways in which he will select the vegetables, considering he will take at least one.
Thus, we get the number of ways as \[16 - 1 = 15\].
Next, let us find the number of ways he can add the salads.
There are three types of salads.
For each of the three salads, he has two choices – whether to add it, or not.
This means that the number of ways in which he can select salads is \[2 \times 2 \times 2 = 8\].
Finally, let us find the number of ways he can add the sauces.
There are two types of sauces.
For each of the two sauces, he has two choices – whether to add it, or not.
This means that the number of ways in which he can select sauces is \[2 \times 2 = 4\].
Now, the number of ways he makes his platter is the product of the number of ways he selects parathas, the number of ways he selects vegetables, the number of ways he selects salads, and the number of ways he selects sauces.
Thus, we get
Number of ways the person makes his platter \[ = 7 \times 15 \times 8 \times 4\] ways
Multiplying the terms, we get
Number of ways the person makes his platter \[ = 3360\] ways

Therefore, there are 3360 ways in which he can make his platter.

Note:
A common mistake is to answer that since there are 3 parathas, and 4 vegetable dishes, he can make the platter in \[ = 4 \times 3 - 1 = 12 - 1 = 11\] ways. This is incorrect.
You should remember that since he prefers to eat paratha and vegetable dishes, he will eat at least one paratha and at least one vegetable dish. Also, remember that for every different item, he has two choices – whether to add it, or not.