Question

In a group of students, $65$ play foot ball, $45$ play hockey, $42$ play cricket, $20$ play foot ball and hockey, $25$ play foot ball and cricket, $15$ play hockey and cricket and $8$ play all the three games. Find the number of students in the group. (Assume that each student in the group plays atleast one game.)

Answer 1

Let ,F Hand C represent the set of students who play foot ball, hockey and cricket respectively. Then n(F)=65, n(H)=45, and n (C)=42.
$n(F\cap H)=20,n(F\cap C)=25,n(H\cap C)=15$ and $n(F\cap H\cap C)=8$
We want to find the number of students in the whole group; that is $n(F\cap H\cap C)$.
By the formula, we have
$n(F\cap H\cap C)=n\left(F\right)+n\left(H\right)+n\left(C\right)-n(F\cap H)-n(H\cap C)-n(F\cap C)+n(F\cap H\cap C)$
$65+45+42-20-25-15+8=100$
Hence, the number of students in the group = 100.