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, we need to find the greatest possible number of students who dislike both pizza and spaghetti.

Let's define:
- \( P \) as the number of students who like pizza.
- \( S \) as the number of students who like spaghetti.
- \( n \) as the total number of students in the class.
- \( x \) as the number of students who like both pizza and spaghetti.
- \( y \) as the number of students who like either pizza or spaghetti or both.
- \( d \) as the number of students who dislike both pizza and spaghetti.

Given:
- \( P = 17 \)
- \( S = 13 \)
- \( n = 25 \)

We know that:

\[
y = P + S - x
\]

Also, the number of students who dislike both pizza and spaghetti is:

\[
d = n - y
\]

To maximize \( d \), we need to minimize \( y \), which means maximizing \( x \).

The maximum value for \( x \) is the smallest value between \( P \) and \( S \), which is:

\[
x_{\text{max}} = \min(P, S) = \min(17, 13) = 13
\]

So, substituting \( x_{\text{max}} \) into the equation for \( y \):

\[
y_{\text{min}} = 17 + 13 - 13 = 17
\]

Finally, the maximum number of students who dislike both pizza and spaghetti is:

\[
d_{\text{max}} = n - y_{\text{min}} = 25 - 17 = 8
\]

Thus, the greatest possible number of students who dislike both pizza and spaghetti is **8**.

Would you like further details or have any questions? Here are some related problems:

1. What is the minimum number of students who dislike both pizza and spaghetti?
2. How many students like only pizza?
3. How many students like only spaghetti?
4. What is the minimum possible number of students who like both pizza and spaghetti?
5. If 3 more students joined the class and all disliked both, what would be the new greatest number of students who dislike both?

**Tip:** Problems involving set theory and Venn diagrams are often easier to solve when you break down the sets and consider intersections and unions carefully.

Discover how to maximize the number of students who dislike both pizza and spaghetti in a class of 25 students, with detailed step-by-step solutions and insights. Explore set theory and Venn diagrams concepts in this math problem suitable for Grades 7-9.

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