Question

In a certain linguistics school there are totally 250 students. Of those 250 students, 40 percent study French as a foreign language, 30 percent study German as a foreign language and 50 percent study Spanish as a foreign language. If 10 students study all these three foreign languages and 10 students didn’t choose these three foreign languages, then how many students are studying in exactly two of these foreign languages?

• A. 20
• B. 30
• C. 40
• D. 50
• E. 60

Hint:

When solving problems involving multiple sets, use the inclusion-exclusion principle to account for overlap.

Answer 1

The Correct Option is C

Step 1: Use the principle of inclusion-exclusion.
Let the total number of students be \( 250 \). We know the number of students studying French, German, and Spanish are 40%, 30%, and 50% of 250, respectively. Therefore:
- French: \( 0.40 \times 250 = 100 \)
- German: \( 0.30 \times 250 = 75 \)
- Spanish: \( 0.50 \times 250 = 125 \)
We are also told that 10 students study all three languages. To find how many students are studying exactly two languages, we will use inclusion-exclusion. First, let's calculate the total number of students who study at least one language: \[ \text{Total students} = 100 + 75 + 125 - \text{students studying two or more languages} \] Since 10 students study all three languages, the number of students studying exactly two languages is 40. Therefore, the correct answer is (C).