To plan the number of meals (breakfast, lunch, and dinner) to be prepared in a cafeteria, a survey was conducted and the following data was obtained: 130 students ate breakfast, 180 students ate lunch, 275 students ate dinner, 68 students ate breakfast and lunch, 112 students ate breakfast and dinner, 90 students ate lunch and dinner, and 58 students at all three meals. How many of the students ate:
(a) [3pts] At least one meal in the cafeteria? (b) [3pts] Exactly one meal in the cafeteria? (c) [3pts] Only dinner in the cafeteria?
(d) [3pts] Exactly two meals in the cafeteria? Hint: Draw a Venn diagram.

To solve this problem, we'll use a Venn diagram and the principle of inclusion and exclusion. Let's denote:
- \(A\) as the set of students who ate breakfast,
- \(B\) as the set of students who ate lunch,
- \(C\) as the set of students who ate dinner.

The given data can be written as:
- \(|A| = 130\)
- \(|B| = 180\)
- \(|C| = 275\)
- \(|A \cap B| = 68\)
- \(|A \cap C| = 112\)
- \(|B \cap C| = 90\)
- \(|A \cap B \cap C| = 58\)

### (a) At least one meal in the cafeteria

Using the principle of inclusion and exclusion:
\[
|A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |A \cap C| - |B \cap C| + |A \cap B \cap C|
\]

Plugging in the values:
\[
|A \cup B \cup C| = 130 + 180 + 275 - 68 - 112 - 90 + 58
\]
\[
|A \cup B \cup C| = 373
\]

So, 373 students ate at least one meal in the cafeteria.

### (b) Exactly one meal in the cafeteria

We can find this by subtracting the number of students who ate two or more meals from the total number who ate each meal:
\[
|A \text{ only}| = |A| - (|A \cap B| + |A \cap C| - |A \cap B \cap C|) = 130 - (68 + 112 - 58) = 130 - 122 = 8
\]
\[
|B \text{ only}| = |B| - (|A \cap B| + |B \cap C| - |A \cap B \cap C|) = 180 - (68 + 90 - 58) = 180 - 100 = 80
\]
\[
|C \text{ only}| = |C| - (|A \cap C| + |B \cap C| - |A \cap B \cap C|) = 275 - (112 + 90 - 58) = 275 - 144 = 131
\]

Adding these values together:
\[
|A \text{ only}| + |B \text{ only}| + |C \text{ only}| = 8 + 80 + 131 = 219
\]

So, 219 students ate exactly one meal in the cafeteria.

### (c) Only dinner in the cafeteria

From the previous calculation for \(C\):
\[
|C \text{ only}| = 131
\]

So, 131 students ate only dinner in the cafeteria.

### (d) Exactly two meals in the cafeteria

We can find this by subtracting the number of students who ate all three meals from those who ate each pair of meals:
\[
|(A \cap B) \text{ only}| = |A \cap B| - |A \cap B \cap C| = 68 - 58 = 10
\]
\[
|(A \cap C) \text{ only}| = |A \cap C| - |A \cap B \cap C| = 112 - 58 = 54
\]
\[
|(B \cap C) \text{ only}| = |B \cap C| - |A \cap B \cap C| = 90 - 58 = 32
\]

Adding these values together:
\[
|(A \cap B) \text{ only}| + |(A \cap C) \text{ only}| + |(B \cap C) \text{ only}| = 10 + 54 + 32 = 96
\]

So, 96 students ate exactly two meals in the cafeteria.

### Summary:
(a) At least one meal in the cafeteria: 373 students
(b) Exactly one meal in the cafeteria: 219 students
(c) Only dinner in the cafeteria: 131 students
(d) Exactly two meals in the cafeteria: 96 students

Would you like any further details or explanations? 

Here are 5 numbered questions you can ask:
1. How do you draw a Venn diagram for three sets?
2. What is the principle of inclusion and exclusion?
3. Can you explain the calculations for exactly one meal in more detail?
4. How do you calculate the number of students who ate only two specific meals?
5. What are other examples of using Venn diagrams in real-life problems?

Tip: When working with Venn diagrams, clearly labeling each section can help avoid confusion and ensure accuracy in your calculations.

Explore how to solve the cafeteria meal distribution problem using Venn diagrams and the principle of inclusion and exclusion. Understand how to calculate the number of students who ate at least one meal, exactly one meal, only dinner, and exactly two meals in the cafeteria. Suitable for high school students.

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