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 find the number of students who like both flavors, we can use the principle of inclusion-exclusion.

Let's define:

- \( C \): the number of students who like chocolate = 15
- \( V \): the number of students who like vanilla = 13
- \( N \): the number of students who like neither = 10
- \( T \): the total number of students = 35
- \( B \): the number of students who like both chocolate and vanilla

We know that the total number of students who like at least one flavor can be calculated as:

\[
\text{Students who like at least one flavor} = T - N = 35 - 10 = 25
\]

Using the principle of inclusion-exclusion for the sets of students who like chocolate and vanilla, we have:

\[
\text{Students who like at least one flavor} = C + V - B
\]

Substituting the known values into this equation, we get:

\[
25 = 15 + 13 - B
\]

Now, simplifying this:

\[
25 = 28 - B
\]

Rearranging gives us:

\[
B = 28 - 25 = 3
\]

Thus, the number of students who like both flavors is \( \boxed{3} \).