Every student of a hostel plays at least one of the games cricket, hockey and football. 45 students play cricket, 38 play hockey and 33 play football. 16 play both cricket and hockey, 12 play both cricket and football, 10 play both hockey and football, and 5 play all three games (each ‘both’ count includes these 5). How many students are there in the hostel?
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 83
Adding the three game totals counts a two-game player twice and a three-game player three times. Taking away the three ‘both’ counts removes the three-game players three times, so they are added back once: \(n=n(C)+n(H)+n(F)-n(C\,H)-n(C\,F)-n(H\,F)+n(C\,H\,F)\).
Sum of the totals \(=45+38+33=116\)
Sum of the ‘both’ counts \(=16+12+10=38\)
Students \(=116-38+5=83\)
Check by regions: cricket only \(=45-16-12+5=22\), hockey only \(=38-16-10+5=17\), football only \(=33-12-10+5=16\); two games only: \(11,\ 7,\ 5\); all three: 5; total \(=22+17+16+11+7+5+5=83\)
Checking the other options
Option (2): 78 — Incorrect. This takes away the three ‘both’ counts but does not add back the 5 who play all three, so they are left out altogether.
Option (3): 116 — Incorrect. This only adds the three game totals, so everyone who plays two or three games is counted more than once.
Option (4): 88 — Incorrect. This adds the 5 who play all three games back twice instead of once.