Question

Paolo's Pizzeria offers 4 types of crust, 6 toppings, and 7 kinds of cheese for the mega calzone. How many different mega calzones can be made if a mega calzone contains 3 different toppings and 2 different cheeses?

Answer 1

Given:

4 types of crust

6 types of toppings

7 kinds of cheese

Find the number of different mega calzones that can be made with 3 different toppings and 2 different kinds of cheese.

Solution:

In choosing a crust, there are only 4 ways to choose since there are only 4 options.

In choosing 3 different toppings out of 6, we can use the combination formula since the order doesn't matter. The formula is:

$nCr=\frac{n!}{r!(n-r)!}$

in which n = 6 and r = 3.

$_6C_3=\frac{6!}{3!(6-3)!}$$_6C_3=\frac{6!}{3!(3!)}$$_6C_3=\frac{6\times5\times4}{3\times2}=\frac{120}{6}=20$

Hence, there are 20 different combinations of three toppings we can form out of 6 available toppings.

Lastly, in choosing 2 kinds of cheese out of 7, we can still use the combination formula in which n = 7 and r = 2.

$_7C_2=\frac{7!}{2!(7-2)!}$$_7C_2=\frac{7!}{2!(5)!}$$_7C_2=\frac{7\times6}{2\times1}=\frac{42}{2}=21$

Hence, there are 21 combinations of 2 kinds of cheese from the 7 available types of cheese.

So, the number of different mega calzones that can be made with 3 different toppings and 2 different kinds of cheese is:

$4crust\times20toppings\times21cheese=1,680$

There are 1, 680 different mega calzones that can be made with 3 different toppings and 2 different kinds of cheese.