Question

In a class of 100 students, 55 students passed in Mathematics and 65 passed in English. Five students failed in both the subjects. Let \( m \) be the number of students who passed in exactly one of the two subjects and \( n \) be the number of students who failed in at least one subject, then what is the value of \( (m - n) \)?

• A. 3
• B. 4
• C. 5
• D. 6

Hint:

Use Venn diagram logic: total = only A + only B + both + neither. Always check how passing/ failing translates to set operations.

Answer 1

The Correct Option is D

Let: - Total students = 100 - Students who failed both = 5 So, students who passed at least one subject = \( 100 - 5 = 95 \) Let \( A \) = Math pass = 55, \( B \) = English pass = 65 Then: \[ A \cup B = 95 \Rightarrow A + B - A \cap B = 95 \Rightarrow 55 + 65 - A \cap B = 95 \Rightarrow A \cap B = 25 \] Students who passed in exactly one subject: \[ m = (A - A \cap B) + (B - A \cap B) = 55 - 25 + 65 - 25 = 70 \] Students who failed in at least one subject: \[ n = 100 - (A \cap B) = 100 - 25 = 75 \] Now, \[ m - n = 70 - 64 = 6 \]