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:

For three-set Venn diagram problems, the formula \( \text{Total Sum of Groups} = (\text{Exactly 1}) + 2 \times (\text{Exactly 2}) + 3 \times (\text{All 3}) \) is very powerful. Combining it with \( \text{Total in at least one} = (\text{Exactly 1}) + (\text{Exactly 2}) + (\text{All 3}) \) allows you to solve for the unknown groups.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This problem involves the principle of inclusion-exclusion for three sets (French, German, and Spanish). We are given the total number of students, the number studying each language, the number studying all three, and the number studying none. We need to find the number of students studying exactly two languages.
Step 2: Detailed Explanation:
First, let's calculate the number of students for each category:

Total students = 250
Number studying French (F) = 40% of 250 = 0.40 \(\times\) 250 = 100
Number studying German (G) = 30% of 250 = 0.30 \(\times\) 250 = 75
Number studying Spanish (S) = 50% of 250 = 0.50 \(\times\) 250 = 125
Number studying all three (F \(\cap\) G \(\cap\) S) = 10
Number studying none of these languages = 10
The total number of students studying at least one language is \(250 - 10 = 240\). Let E1 be the number of students studying exactly one language, E2 be the number studying exactly two, and E3 be the number studying all three. We know: \[ \text{Total studying} = E1 + E2 + E3 \] \[ 240 = E1 + E2 + 10 \implies E1 + E2 = 230 \] We also use the formula that relates the sum of individual sets to these groups: \[ N(F) + N(G) + N(S) = E1 + 2 \times E2 + 3 \times E3 \] Substituting the known values: \[ 100 + 75 + 125 = E1 + 2 \times E2 + 3 \times 10 \] \[ 300 = E1 + 2 \times E2 + 30 \] \[ E1 + 2 \times E2 = 270 \] Now we have a system of two linear equations: 1) \(E1 + E2 = 230\) 2) \(E1 + 2 \times E2 = 270\) Subtract equation (1) from equation (2): \[ (E1 + 2 \times E2) - (E1 + E2) = 270 - 230 \] \[ E2 = 40 \] Step 3: Final Answer:
Based on a rigorous application of set theory principles, the number of students studying exactly two languages is 40.