Question

In a group of students, 10 students like Mathematics, 12 students like English, 4 students like both Mathematics and English, and 6 students like neither Mathematics nor English. The number of students in the group is

• A. 18
• B. 20
• C. 24
• D. 32

Hint:

Use the formula for the union of two sets, then add the students who like neither subject.

Answer 1

The Correct Option is C

Instead of applying the union formula directly, build the answer by filling in each region of a two-circle Venn diagram, one region at a time, then add every region together.

1. Both Mathematics and English: This is the overlap of the two circles, given directly as $4$ students.
2. Only Mathematics: Out of the $10$ students who like Mathematics, $4$ of them are already counted in the overlap, so the remaining $10 - 4 = 6$ students like Mathematics only.
3. Only English: Out of the $12$ students who like English, $4$ are in the overlap, so the remaining $12 - 4 = 8$ students like English only.
4. Neither subject: Given directly as $6$ students, sitting outside both circles.

Now add up all four separate, non-overlapping regions to get the full group: only Mathematics ($6$) plus only English ($8$) plus both ($4$) plus neither ($6$):

\[ 6 + 8 + 4 + 6 = 24 \]

Let's summarize:

• Splitting the group into "only Mathematics", "only English", "both", and "neither" avoids any risk of double-counting the $4$ overlap students.
• Adding all four non-overlapping regions gives the true total group size.

So the number of students in the group is $24$, matching option (C).