Question

Out of 280 students in Class XII of a school, 135 play hockey, 110 play football, 80 play volleyball, 35 of these play hockey and football, 30 play volleyball and hockey, 20 play football and volleyball. Also each student plays at least one of the three games. How many students play all.

Answer 1

Let all three play n
arq
280=135+110+80-35-30-20-n
n=40

Answer 2

Given:

Total students (N) = 280

Total students play hockey = 135

Total students play football = 110

Total students play volleyball = 80

Total students play hockey and football = 35

Total students plays volleyball and hockey = 30

Total students play football and volleyball = 20

To Find:

The number of students play all the games =  n(F∩H∩V)

Solution:

As given,

n(H) = 135

n(F) = 110

n(V) = 80

n(H∩F) = 30

n(F∩V) = 20

n(H∩F) = 35

And,

n(F∪H∪V) = 280

we know ,

n(F∪H∪V) = n(H) + n(F) + n(V) - n(H∩F) - n(F∩V) - n(H∩F) + n(F∩H∩V)

280 = 135 + 110 + 80 - 35 -20 - 30 +  n(F∩H∩V)

280 = 325 - 85 +  n(F∩H∩V)

280 = 240 +  n(F∩H∩V)

280 -240 =  n(F∩H∩V)

40 =  n(F∩H∩V)

Hence, the number of students play all sports are 40.