Question

In a group of 240 student. 144 plays football; 130 play tennis, 106 play hockey. if 20 of the students plays both football and tennis, 60 plays football and hockey, 42 play tennis and hockey and each student play at least one of the three game. find the number of students that played all games​

Answer 1

Total number of students playing football (F): 144

Total number of students playing tennis (T): 130

Total number of students playing hockey (H): 106

Number of students playing both football and tennis (F ∩ T): 20

Number of students playing both football and hockey (F ∩ H): 60

Number of students playing both tennis and hockey (T ∩ H): 42

calculation of the total number of students who play at least one of the three games

Total number of students playing at least one game = F + T + H - (F ∩ T) - (F ∩ H) - (T ∩ H) + (F ∩ T ∩ H)

Total number of students playing at least one game = 144 + 130 + 106 - 20 - 60 - 42 + (F ∩ T ∩ H)

the total number of students is 240, therefore:

240 = 144 + 130 + 106 - 20 - 60 - 42 + (F ∩ T ∩ H)

we solve for (F ∩ T ∩ H):

240 = 258 - 122 + (F ∩ T ∩ H)

122 + (F ∩ T ∩ H) = 258 - 240

(F ∩ T ∩ H) = 18

18 students played all three games: football, tennis and hockey.

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