Venn diagram question-  18 liked pizza, 10 liked burgers, 10 liked sushi, 7 liked burgers and pizza, 5 liked burgers and sushi, 2 liked pizza and sushi, and 3 liked all three. How many people like exactly two types of food?

We can solve this using a Venn diagram approach, applying the principle of inclusion and exclusion. The goal is to find how many people like exactly two types of food.

### Given:
- \( P \) (Pizza) = 18 people like pizza
- \( B \) (Burgers) = 10 people like burgers
- \( S \) (Sushi) = 10 people like sushi
- \( P \cap B \) (Pizza and Burgers) = 7 people like both pizza and burgers
- \( B \cap S \) (Burgers and Sushi) = 5 people like both burgers and sushi
- \( P \cap S \) (Pizza and Sushi) = 2 people like both pizza and sushi
- \( P \cap B \cap S \) (All three) = 3 people like all three foods

### To find the number of people who like exactly two types of food:

We know the following:
1. The number of people who like both pizza and burgers includes those who like all three foods. Therefore, people who like **exactly** pizza and burgers are:
   \[
   (P \cap B) - (P \cap B \cap S) = 7 - 3 = 4
   \]
2. Similarly, the number of people who like **exactly** burgers and sushi is:
   \[
   (B \cap S) - (P \cap B \cap S) = 5 - 3 = 2
   \]
3. The number of people who like **exactly** pizza and sushi is:
   \[
   (P \cap S) - (P \cap B \cap S) = 2 - 3 = -1
   \]
   However, this negative value indicates that all people who like pizza and sushi also like burgers. Therefore, no one likes exactly pizza and sushi.

### Total number of people who like exactly two types of food:
\[
4 \text{ (Pizza and Burgers)} + 2 \text{ (Burgers and Sushi)} + 0 \text{ (Pizza and Sushi)} = 6
\]

Thus, **6 people like exactly two types of food.**

Would you like a breakdown of other relationships, or have any questions?

### Related questions:
1. How many people like only pizza?
2. How many people like all three types of food?
3. How many people like only one type of food?
4. What is the total number of people surveyed?
5. How many people like exactly one type of food?

### Tip:
When dealing with Venn diagrams, using the principle of inclusion-exclusion can simplify finding overlapping and distinct sets.

Venn diagram question - 18 liked pizza, 10 liked burgers, 10 liked sushi, 7 liked burgers and pizza, 5 liked burgers and sushi, 2 liked pizza and sushi, and 3 liked all three. How many people like exactly two types of food?

Solve a Venn diagram problem involving pizza, burgers, and sushi. Learn how to apply the inclusion-exclusion principle to determine how many people like exactly two types of food. This problem uses set theory concepts and is suitable for students in Grades 7-10.

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