Question

A pizza place offers 4 different cheeses and 10 different toppings. In how many ways can a pizza be made with 2 cheese and 3 toppings?

Answer 1

Hint: This is the question of combinations. We know that combination can be defined as the way of selecting items from a collection in such a way that the order of selection does not matter. Therefore, in this problem, it is not important in what order the toppings and cheeses are put on the pizza. So we will use the combination formula to solve this problem.
Formula used:
 $ {C_{n,k}} = \dfrac{{n!}}{{k!(n - k)!}} $ , where $ n $ is the population and $ k $ is the picks.

Complete step-by-step answer:
Let us first consider the case of the cheese. We are given that the pizza place has four different types of cheeses. And from these, we need to choose two.
Therefore, we have the population $ n = 4 $ and picks $ k = 2 $ .
Now, we will use the formula $ {C_{n,k}} = \dfrac{{n!}}{{k!(n - k)!}} $ .
 $ \Rightarrow {C_{4,2}} = \dfrac{{4!}}{{2!(4 - 2)!}} = \dfrac{{4!}}{{2!2!}} = \dfrac{{24}}{{2 \times 2}} = 6 $
Now, we will do the same thing for toppings. We are given that the pizza place has ten different types of toppings. And from these, we need to choose three.
Therefore, we have the population $ n = 10 $ and pick $ k = 3 $ .
Now, we will use the formula $ {C_{n,k}} = \dfrac{{n!}}{{k!(n - k)!}} $ .
 $ \Rightarrow {C_{10,3}} = \dfrac{{10!}}{{3!(10 - 3)!}} = \dfrac{{10!}}{{3!7!}} = \dfrac{{10 \times 9 \times 8 \times 7!}}{{6 \times 7!}} = 120 $
Now, we have to make pizza with 2 cheeses and 3 toppings together and these we can choose individually. Therefore, we can get our final answer by multiplying these two together,
 $ \Rightarrow 6 \times 120 = 720 $
Thus, there are 720 different ways of making pizza with 2 cheese and 3 toppings.
So, the correct answer is “720”.

Note: Here, in this question, we have used the concept of factorial. The factorial function can be defined as the multiplication of all whole numbers from our chosen number down to 1. For example, $ 7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 $ .