Question

In a class of 100 students, 55 students passed in Mathematics and 65 passed in English. Five students failed in both subjects. Let \( n \) be the number of students who passed in exactly one of the two subjects and \( m \) 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. 7

Hint:

In set theory problems involving inclusion and exclusion, carefully apply the formula for union and intersection to calculate the number of students in various categories.

Answer 1

The Correct Option is C

We are given:
- Total students = 100
- Students passing in Mathematics = 55
- Students passing in English = 65
- Students failing in both subjects = 5
Let \( A \) be the set of students passing Mathematics and \( B \) be the set of students passing English.
We are asked to find \( (m - n) \), where:
- \( m \) is the number of students failing at least one subject.
- \( n \) is the number of students passing exactly one subject.
To find \( n \) and \( m \), we use the principle of inclusion and exclusion.
Let \( x \) be the number of students passing both subjects.
We know: \[ |A| = 55, \quad |B| = 65, \quad |A \cap B| = x \] By the principle of inclusion and exclusion: \[ |A \cup B| = |A| + |B| - |A \cap B| = 55 + 65 - x = 120 - x \] Also, the number of students failing at least one subject is: \[ m = 100 - |A \cup B| = 100 - (120 - x) = x - 20 \] The number of students passing exactly one subject is: \[ n = (|A| - |A \cap B|) + (|B| - |A \cap B|) = (55 - x) + (65 - x) = 120 - 2x \] Now, \( m - n = (x - 20) - (120 - 2x) = x - 20 - 120 + 2x = 3x - 140 \).
For \( x = 45 \), the value of \( m - n = 5 \).
Thus, the value of \( (m - n) \) is 5.