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

Step 1: Understanding the Question.
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. We need the total number of students in the whole group.

Step 2: Key Formula or Approach.
The number of students who like at least one of the two subjects (Mathematics or English) is given by the inclusion-exclusion formula:
\[ |M \cup E| = |M| + |E| - |M \cap E| \]
This formula subtracts the overlap once, since students who like both subjects would otherwise be counted twice, once in \(|M|\) and once in \(|E|\).

Step 3: Detailed Explanation.
Substitute the given values:
\[ |M \cup E| = 10 + 12 - 4 = 18 \]
So \(18\) students like at least one of Mathematics or English. The total group also includes the 6 students who like neither subject, and these students are completely separate from the 18 counted above, since they are outside both \(M\) and \(E\). Adding them gives the full group size:
\[ \text{Total students} = |M \cup E| + \text{neither} = 18 + 6 = 24 \]

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