In a class of 84 students, 55 play cricket, 47 play football and 42 play volleyball. 20 play both cricket and football, 22 play both football and volleyball and 26 play both cricket and volleyball. If each student plays at least one game, find the number of students who play only cricket.?

Option 3 : 17

Calculation:

Let C, F and V denote the sets of number of students who play cricket, football and volleyball, respectively.

⇒ n(C) = 55, n(F) = 47 and n(V) = 42

⇒ n(C ∩ F) = 20, n(F ∩ V) = 22, n(C ∩ V) = 26

⇒ Let 'x' be the number of students who play all the 3 games.

⇒ Number of students who play both cricket and football, but not volleyball = (20 - x)

⇒ Similarly, number of students who play both football and volleyball, but not cricket = (22 - x)

⇒ Number of students who play both cricket and volleyball, but not football = (26 - x) 

⇒ Now, we can find the number of students who play cricket only, football only and volleyball only as follows:

⇒ n(C) only = 55 - (20 - x + x + 26 - x) = x + 9

⇒ Similarly for n(V) only = x + 5

⇒ And n(F) only =  = x - 6 

⇒ 84 = (x + 9) + (x + 5) + (x – 6) + (20 - x) + (26 - x) + (22 - 4) + x

⇒ x = 8

⇒ Number of students who play only cricket = 8 + 9 = 17