ITPEC FE Morning October 2023 Question 1
In a school class, 13 students play basketball and 19 play football. Among them, 8 students play both. Further, 7 students play neither. How many students are there in the class?
Explanation
31 students — count the students who play at least one sport with inclusion–exclusion, then add those who play neither.
- •At least one sport: 13 + 19 − 8 = 24; the 8 who play both are counted once in 13 and again in 19, so they are subtracted once
- •Whole class: 24 + 7 = 31
Check by regions: basketball only 13 − 8 = 5, football only 19 − 8 = 11, both 8, neither 7, and 5 + 11 + 8 + 7 = 31.
Answer (b)
Why not others:
- •(a) 17 — subtracts the 7 who play neither, 24 − 7, when they must be added to the class
- •(c) 33 — matches no consistent count; the four disjoint groups (5, 11, 8, 7) sum to 31
- •(d) 47 — adds all four numbers, 13 + 19 + 8 + 7, counting the 8 who play both three times
Key rule: |A ∪ B| = |A| + |B| − |A ∩ B|; the people in neither set lie outside the union and are added afterwards.
Basic Theory · AI-written, may contain errors
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