Q.4
25 persons are in a room. 15 of them play hockey, 17 of them play football and
10 of them play both hockey and football. Then the number of persons playing
neither hockey nor football is:
- 2
- 17
- 13
- 3
The number of persons playing neither hockey nor football is 3.
Problem Breakdown
Total persons: 25. Hockey players (H): 15. Football players (F): 17. Both (H ∩ F): 10.
Use inclusion-exclusion: |H ∪ F| = |H| + |F| – |H ∩ F| = 15 + 17 – 10 = 22.
Neither: Total – |H ∪ F| = 25 – 22 = 3.
Venn Diagram Regions
-
Only hockey: 15 – 10 = 5
-
Only football: 17 – 10 = 7
-
Both: 10
-
Total playing at least one: 5 + 7 + 10 = 22
-
Neither: 25 – 22 = 3
Option Analysis
-
2: Too low; undercounts by 1, perhaps from misadding only regions as 5 + 7 = 12, then 25 – 23 = 2.
-
17: Equals football players; ignores hockey overlap and total constraint.
-
13: 25 – 12 (only hockey + only football, forgetting both); double-counts intersection.
-
3: Correct, as derived from precise inclusion-exclusion principle.
This set theory problem appears in competitive exams like GATE, testing Venn diagrams and unions for quick solving.
