In a class of 25 students, 17 like pizza and 13 like spaghetti. What is the greatest possible number of students who dislike both pizza and spaghetti (note that some may not like either)?

To solve this problem, let's denote:

- \( n = 25 \): The total number of students.
- \( n(P) = 17 \): The number of students who like pizza.
- \( n(S) = 13 \): The number of students who like spaghetti.

We need to find the greatest possible number of students who dislike both pizza and spaghetti. Let's use the principle of inclusion-exclusion:

\[
n(P \cup S) = n(P) + n(S) - n(P \cap S)
\]

Where \( n(P \cup S) \) is the number of students who like either pizza or spaghetti or both, and \( n(P \cap S) \) is the number of students who like both.

The greatest number of students who dislike both pizza and spaghetti would be when the number of students who like both \( n(P \cap S) \) is minimized. 

The smallest possible value for \( n(P \cap S) \) is when as few students as possible like both, which would be when:

\[
n(P \cap S) = \max(0, n(P) + n(S) - n) = \max(0, 17 + 13 - 25) = \max(0, 5) = 5
\]

So,

\[
n(P \cup S) = 17 + 13 - 5 = 25
\]

However, if \( n(P \cap S) \) is 5, then all students like at least one of pizza or spaghetti, meaning no student dislikes both. But, if we increase \( n(P \cap S) \) beyond 5, it will reduce \( n(P \cup S) \), thereby increasing the number of students who dislike both.

Therefore, the greatest number of students who dislike both would occur if the minimum number of students like both. Thus,

\[
\text{Greatest number of students who dislike both} = n - n(P \cup S) = 25 - 25 = 0
\]

**Final Answer**: The greatest possible number of students who dislike both pizza and spaghetti is **0**.

Would you like more details or have any questions?

Here are some related questions to explore:

1. If 5 students like both pizza and spaghetti, how many like only pizza?
2. What is the minimum number of students who dislike both pizza and spaghetti?
3. How would the answer change if the class size were increased to 30 students?
4. What is the maximum number of students who like only pizza?
5. How many students like exactly one of the two foods?

**Tip**: When solving problems involving set operations, the principle of inclusion-exclusion is a powerful tool to calculate unions and intersections accurately.

Learn how to solve a problem involving students liking pizza and spaghetti with a detailed step-by-step solution. Understand the principle of inclusion-exclusion in set theory to find the greatest number of students who dislike both foods. Suitable for students in Grades 7-9.

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