Question

A restaurant offered a choice of $4$ salads, $9$ main courses and $3$ desserts. How many possible $3-$course meals are there?

Answer 1

Hint: We will use the combination here. A $3-$course meal contains $1$ salad, $1$ main course and $1$ dessert. We will calculate how many of the courses can be chosen from each one of these courses using the combination.

Complete step by step answer:
Let us consider the given problem.
We are asked to find the number of $3-$course meals there are.
To find the number, we will first consider each of the courses separately.
Let us say that a $3-$course meal contains a salad, a main course and a dessert.
So, we have to cho0se one from each of the courses.
We know that there are $4$ salads. So, we have to choose one out of these to insert in a $3-$course meal. And we can choose any one of these $4$ salads. So, using combination, we will get ${}^{n}{{C}_{r}}={}^{4}{{C}_{1}}=\dfrac{4}{1}=4.$
Similarly, we need to choose one of the $9$ main courses. We will get ${}^{n}{{C}_{r}}={}^{9}{{C}_{1}}=\dfrac{9}{1}=9.$
In the same way, we need to choose one from the $3$ desserts. We will get ${}^{n}{{C}_{r}}={}^{3}{{C}_{1}}=\dfrac{3}{1}=3.$
We need to multiply these values to get the number of $3-$course meals there are.

Hence the number of $3-$course meal is ${}^{4}{{C}_{1}}{}^{9}{{C}_{1}}{}^{3}{{C}_{1}}=4\times 9\times 3=108.$

Note: The combination is a unique way to arrange a number of objects. If we have $n$ number of objects and we need to choose any $r$ of them, $r