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A group of 60 students contains 23 who play hockey, 15 who play basketball, and 20 who play cricket. Also, 7 students play both hockey and basketball, 5 play both cricket and basketball, 4 play both hockey and cricket, and 15 students do not play any of these three games. Which statement is true?

1. 4 students play hockey, basketball, and cricket
2. 20 students play hockey but not cricket
3. 1 student plays hockey and cricket but not basketball
4. All of the above are correct

Correct answer: 1 student plays hockey and cricket but not basketball

Solution

Union = 60 - 15 = 45. By inclusion-exclusion 45 = 23+15+20 -7-5-4 + n(all), giving n(all)=3. Then hockey-and-cricket-only = 4 - 3 = 1, so exactly 1 student plays hockey and cricket but not basketball.

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