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 inclusion-exclusion for the two subjects, then add the students who like neither: (10 + 12 - 4) + 6.

Answer 1

The Correct Option is C

Step 1: Set up the sets.
Let M be the set of students who like Mathematics and E be the set of students who like English. We are given \(|M| = 10\), \(|E| = 12\), \(|M \cap E| = 4\) (students who like both), and 6 students who like neither Mathematics nor English at all.

Step 2: Find the number of students who like at least one of the two subjects.
Use the inclusion-exclusion rule for two sets, which avoids double-counting the students who like both subjects:
\[ |M \cup E| = |M| + |E| - |M \cap E| \]
\[ |M \cup E| = 10 + 12 - 4 = 18 \]
So 18 students like Mathematics or English (or both).

Step 3: Add the students who like neither subject.
The 6 students who like neither Mathematics nor English are outside the set \(M \cup E\) entirely, so they must be added separately to get the full group size:
\[ \text{Total} = |M \cup E| + \text{neither} = 18 + 6 = 24 \]

Step 4: Final Answer.
The total number of students in the group is 24, which matches option (C). \[ \boxed{24} \]