In a group of students, $65$ play football, $45$ play hockey, $42$ play cricket, $20$ play football and hockey, $25$ play football and cricket, $15$ play hockey and cricket, and 8 play all three games. Find the total number of students in the group.

(Assume that each student in the group plays at least one game.)

The correct option is A: $100$

Let F, H and C represent the set of students who play football, hockey and cricket respectively.

From the given information, we have

$n\left(F\right)=65,$ $n\left(H\right)=45,$ $n\left(C\right)=42,$

$n(F\cap H)=20,$ $n(F\cap C)=25,$ $n(H\cap C)=15$

$n(F\cap H\cap C)=8$

From the above diagram,we have

Total number of students in the group

$=28+12+18+7+10+17+8$

$=100$
Hence, the total number of students in the group is $100.$