Question: In a class of 60 students, 23 play hockey, 15 play basketball, 20 play cricket and 7 play hockey and basketball,5 play cricket and basketball,4 play hockey and cricket,15 do not play any of the three games. Find how many play hockey, basketball and cricket.

(A) 3
(B) 5
(C) 4
(D) 2
(E) 1

Correct Answer: (A)
Approach Solution : 1

There are 15 students who do not play any of three games.
n(H ∪ B ∪ C) = 60 – 15 = 45
So we can write that,
=> n(H ∪ B ∪ C) = n(H) + n(B) + n(C) – n(H ∩ B) – n(B ∩ C) – n(C ∩ H) + n(H ∩ B ∩ C)
Let the number of student who play all games, that is n(H ∩ B ∩ C) be x.
=> 45 = 23 + 15 + 20 – 7 – 5 – 4 + x
=> 45 = 42 + x
=> x = 45- 42 = 3
Number of students who play all the three games = 3

Approach Solution : 2

The information given in the question is detailed below.
Number of students, n ( Total) = 60
Number of hockey players, n(H) = 23
Number of basketball players, n (B) = 15
Number of cricket players, n(C) = 20
Number of students playing both hockey and basketball, n ( H ∩ B) = 7
Number of students playing hockey and cricket, n ( H ∩ C) = 4
Number of students playing cricket and basketball, n ( C ∩ B) = 5
Number of students who do not play anything, n (none) = 15
We have to find the number of students playing all games, n ( H ∩ B ∩ C ).
We know that,
n ( Total) = n(H) + n (B) + n(C) - n ( H ∩ B) - n ( H ∩ C) - n ( C ∩ B) + n ( H ∩ B ∩ C ) + n (none)
=> 60 = 23 + 15 + 20 - 7 - 4 - 5 + n ( H ∩ B ∩ C ) + 15
=> n ( H ∩ B ∩ C ) = 3
As a result, there are students who play hockey, basketball and cricket together.

“In a class of 60 students,23 play hockey,15 play basketball” - is a topic that is covered in the quantitative reasoning section of the GMAT. To successfully execute the GMAT Problem Solving questions, a student must possess a wide range of qualitative skills. The entire GMAT Quant section consists of 31 questions. The problem-solving section of the GMAT Quant topics requires the solution of calculative mathematical problems.

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