Question

In a school where there was a compulsion to learn at least one foreign language from the choice given to them, namely German, French and Spanish. Twenty eight students took French, thirty took German and thirty two took Spanish. Six students learnt French and German, eight students learnt German and Spanish, ten students learnt French and Spanish. Fifty four students learnt only one foreign language while twenty students learnt only German. Find the number of students in the school.

• A. 60
• B. 62
• C. 70
• D. none of the above

Answer 1

The Correct Option is C

To solve the problem, we use the principle of inclusion-exclusion and Venn diagrams. We have the sets:

F (French) = 28, G (German) = 30, S (Spanish) = 32. Intersections given are: |F∩G| = 6, |G∩S| = 8, |F∩S| = 10.

• So, |F∩G∩S| needs to be found. 
• Using inclusion-exclusion for total students, |F ∪ G ∪ S| is given by:

|F ∪ G ∪ S| = |F| + |G| + |S| - |F∩G| - |G∩S| - |F∩S| + |F∩G∩S|

Substituting the known values:

• |F ∪ G ∪ S| = 28 + 30 + 32 - 6 - 8 - 10 + |F∩G∩S| = 66 + |F∩G∩S|

Given: 54 students learn only one language, and 20 students learn only German.

• Let x be the number of students who learn all three languages.
• 54 = Students learning only one language implies: |F only| + |G only| + |S only| = 54.

From |G only| = 20, we need:

• From the equation for |G|:
• |G only| = |G| - |F∩G| - |G∩S| + |F∩G∩S| = 20.

20 = 30 - 6 - 8 + x

Solving: x = 4. Now substitute x = 4 back in the equation:

• |F ∪ G ∪ S| = 66 + 4 = 70.

Languages	Students
Only French	|F| - |F∩G| - |F∩S| + |F∩G∩S| = 10
Only German	20
Only Spanish	|S| - |G∩S| - |F∩S| + |F∩G∩S| = 24
French & German	6
German & Spanish	8
French & Spanish	10
All three languages	4

Therefore, the number of students in the school is 70. Thus, the correct answer is 70.