Question

In a class, there are 15 students who like chocolate. 13 students like vanilla. 10 students like neither. If there are 35 people in the class, how many students like both flavors?

Answer 1

To solve this problem, we can use a Venn diagram approach, but let's break it down step by step using set notation.

Let:
- \( A \) be the set of students who like chocolate.
- \( B \) be the set of students who like vanilla.
- \( |A| = 15 \) (students who like chocolate).
- \( |B| = 13 \) (students who like vanilla).
- \( |A^c \cap B^c| = 10 \) (students who like neither).

The total number of students in the class is:
\[
|U| = 35
\]

We want to find the number of students who like both flavors, which we will denote as \( |A \cap B| \).

First, we can find the number of students who like either or both flavors. This is given by the total minus those who like neither:
\[
|A \cup B| = |U| - |A^c \cap B^c| = 35 - 10 = 25
\]

Now, we can apply the principle of inclusion-exclusion to find \( |A \cap B| \):
\[
|A \cup B| = |A| + |B| - |A \cap B|
\]
Substituting in the known values:
\[
25 = 15 + 13 - |A \cap B|
\]
This simplifies to:
\[
25 = 28 - |A \cap B|
\]
Rearranging this gives:
\[
|A \cap B| = 28 - 25 = 3
\]

Thus, the number of students who like both flavors is:
\[
\boxed{3}
\]