Discover how to count all the topping combinations on a pizza

Pizza is a great food.  It’s an entire meal with all the food groups in one slice.  But there is more that can be learned from a pizza.  Will all the different toppings and types of crusts a pizza can be with there are many different combinations of ingredients and thus pizzas that can be made.  A pizza is traditionally made in the shape of a circle.  Usually when you order a pizza, you order by size, small, medium, large and sometimes extra large.  These size descriptions are not very informative because how much pizza are you getting when you order a small?  Once you know the radius or diameter of the pizza, then we can start calculating how much pizza we’re really getting for our dime!

What are combinations?

_nC_r = \frac{n!}{(n-r)! r!} 

This means, if we are given n objects and we only want to choose r of these objects and the order in which we choose them doesn’t matter, how many ways can we choose r of the n objects?  We are assuming that n \geq r  . 

What does n! mean?

n! is the “factorial” of n or “n factorial”.   This number is equal to the number of ways of selecting all n objects one at a time where the order of selection matters and the object is not replaced.  To elaborate, there are n ways to choose the first object; (n-1) to choose the second object; (n-2) ways to the third object and so on until there is only 1 object left to choose.  The total number of ways of selecting all these n objects, where order matters and there is no replacements is equal to n(n-1)(n-2) \cdots (2)(1) = n! 

Combinations of pizza

Below are the list of toppings and crust options from the Gabriel pizza menu to create your own pizza.

Let’s put a few restrictions on the creation of a pizza, for example, you can only choose one crust type and one crust thickness; no more than 5 veggies; no more than 3 meats; no more than 3 cheeses.  So how many different types of pizza can we make from the ingredients and options below?   This is where combinations come into play. 

Let’s have a look at the menu below. How many options for crust thickness; how many options for crust type are there; how many vegetable toppings are there; how many meat topping options are there; how many cheese options are there?

Crust thickness = 3

Crust type = 3

Veggies = 24

Meat = 14

Cheese = 7

Let’s say our first pizza must have the following number of each type of items:

1 crust thickness

1 crust type

5 vegetable toppings

2 meat toppings

1 cheese topping

The question is: 

How many possible combinations of pizza are there with these restrictions?

Solution:

How many ways can we choose the crust thickness?

  _3C_1 = \frac{3!}{2! 1!} = 3

How many ways can we choose the crust type?

  _3C_1 = \frac{3!}{2! 1!} = 3

How many ways can we choose 5 vegetable toppings?

  _24C_5 = \frac{24!}{19! 5!} = 42504

How many ways can we choose 2 meet toppings?

  _14C_2 = \frac{14!}{12! 2!} = 91

And finally, how many ways can we choose 1 cheese?

  _7C_1 = \frac{7!}{6! 1!} = 7

Now to find the total number of possible pizzas with this combination of toppings can be found by multiplying all these above numbers together.

  _3 \times 3 \times 42504 \times 91 \times 7 = 243 675 432 is the number of possible pizza combinations that can be made with the imposed restrictions on toppings.  That’s a lot of different pizzas!

Exercises:

1. How many cheese pizzas can be made if we are told that a cheese pizza must have at least 5 different cheese on it and no other toppings?
2. How many meat pizzas can be made if we are told that a meat pizza must have at least 5 different types of meat on it, 1 cheese, and no other toppings?
3. How many different vegetarian pizzas can be made if we are told that a vegetarian pizzas has at least 5 different types of vegetables on it, 1 cheese and no other toppings?
4. How many different combination pizzas can be made if we are told a combination pizzas has 6 different meats, 6 different vegetables and 3 different cheeses?
5. How many different pepperoni pizzas are there? ( A pepperoni pizza is a pizza with only pepperoni and cheese on it.)
6. How many thin crust, gluten free vegetarian pizzas are possible?
7. How many thick crust, whole wheat flour crust meat pizzas are there?
8. How many regular thick, regular flour crust cheese pizzas are possible?
9. Suppose Joe only likes pizzas that have both pineapple and hot peppers. How many possible pizzas can Joe choose from?
10. Mary is a very picky eater and doesn’t like olives on her pizza; she hates any vegetable that is green; she won’t eat pork of any kind; she hates Feta cheese, goat cheese and ricotta cheese, and will only eat thin crust pizzas. How many pizzas can picky Mary choose from?