Explanation:

Let's break down the question:

- Total students enrolled: 180

- Students studying French: 80

- German: 90

- Chinese: 100

- Students studying more than one subject = 50% more than those studying all three.

- Let the number studying all three = x.

- Then, students studying >1 language = x + 0.5x = 1.5x.

By inclusion-exclusion principle:

Number with at least one language = (80 + 90 + 100) - (students in exactly 2 languages) - 2x

But we don't know students in exactly 2 directly.

Given:

Total = those in only one + those in exactly two + those in all three

So:

180 = (80 + 90 + 100) - (number in exactly two languages) - 2x

This simplifies to:

180 = 270 - (number in exactly two languages) - 2x

number in exactly two languages + 2x = 90

But total in more than one language includes exactly two and all three:

number in exactly two + x = 1.5x ? number in exactly two = 0.5x

So

0.5x + 2x = 90 ? 2.5x = 90 ? x = 36

- Option:1, 18

- Option:2, 24

- Option:3, 36

- Option:4, 42

Answer:

- The correct option is Option:3, 36.

- Number studying all three = 36.

- Students in more than one language = 1.5 x 36 = 54.

- The inclusion-exclusion principle confirms this calculation.

- Other options (18, 24, 42) do not satisfy all conditions.