# One weekend, there were 77 pizzas ordered for the sophomore dorm. That weekend 19 customers ordered onions on their pizza, 15 customers ordered their pizza with just pepperoni, 18 customers...

One weekend, there were 77 pizzas ordered for the sophomore dorm. That weekend 19 customers ordered onions on their pizza, 15 customers ordered their pizza with just pepperoni, 18 customers ordered their pizza with just sausage, 9 ordered theirs with pepperoni and sausage, 8 ordered theirs with sausage and onions, 5 ordered theirs with pepperoni and onions, and 4 ordered theirs with all three items. The remaining pizzas were cheese pizzas with no toppings. Draw a Venn diagram and answer the following questions. How many customers ordered at most two toppings on their pizza? How many customers ordered their pizza with pepperoni or sausage?

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### 1 Answer

First, let us identify all possible events. There are three possible toppings - onions, pepperoni, or sausage; and there's an option to add no topping at all.

Hence, possible orders are:

- No Topping
- Onion Only
- Pepperoni Only
- Sausage Only
- Onion and Pepperoni Only
- Onion and Sausage Only
- Pepperoni and Sausage Only
- Onion, Pepperoni and Sausage

Attached is a Venn Diagram. The three circles represent Pepperoni Pizzas, Onion Pizzas, and Sausage Pizzas. The area outside the circles represent Cheese Pizzas (no topping).

We start filling out the diagram using single-topping pizzas (e.g. pepperoni only) and full-topping pizzas (i.e. all toppings, intersection of the three circles).

How many people ordered at most two toppings? For this question, we want to count the total number of people who order a Cheese Pizza (0 toppings), a 1-topping pizza and a 2-topping pizza -- in short, all orders except for the 3-topping pizza. The 3-topping pizza is the intersection of all three circles. As you can see from the diagram, there are 4 such orders. Hence, 77 - 4 = 73 people ordered at most two toppings with their pizzas.

How many order pizzas with Pepperoni or Sausage? Here, we are concerned with the union of Pepperoni and Sausage. That is, all the numbers inside the circle of Pepperoni or Sausage. In this case, that would be 15 (Pepperoni Only) + 1 (Pepperoni + Onion Only) + 4 (All Toppings) + 5 (Pepperoni + Sausage Only) + 4 (Sausage + Onion Only) + 18 (Sausage Only). These are orders that contain either Sausage OR Pepperoni. The total is 47 orders.