Question

In a class of 80 students numbered 1 to 80, all odd numbered students opt for Cricket, students divisible by 5 opt for Football, and those divisible by 7 opt for Hockey. The number of students who do not opt any of the three games is

• A. \(13\)
• B. \(24\)
• C. \(28\)
• D. \(52\)
• E. \(67\)

Hint:

Use Inclusion-Exclusion carefully and always subtract overlaps before adding triple intersections.

Answer 1

The Correct Option is C

Understanding the Concept: We use the Principle of Inclusion-Exclusion: \[ n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(B \cap C) - n(C \cap A) + n(A \cap B \cap C) \]

Step 1: Count individual sets
Odd numbers (Cricket): \[ 40 \] Multiples of 5 (Football): \[ \lfloor 80/5 \rfloor = 16 \] Multiples of 7 (Hockey): \[ \lfloor 80/7 \rfloor = 11 \]

Step 2: Count pairwise intersections
Odd and divisible by 5: multiples of 5 that are odd → 5,15,...75 → 8 numbers Multiples of 5 and 7: LCM = 35 → \( \lfloor 80/35 \rfloor = 2 \) Odd and divisible by 7: odd multiples of 7 → 7,21,...77 → 6 numbers

Step 3: Triple intersection
Odd multiple of 35: 35 only → 1

Step 4: Apply formula
\[ n = 40 + 16 + 11 - 8 - 2 - 6 + 1 = 52 \]

Step 5: Students not choosing any
\[ 80 - 52 = 28 \] \[ \boxed{28} \]