Question

Of the members of three athletic team s in a school $21$ are in cricket team, $26$ are in hockey team and $29$ are in the football team. Among them, $14$ play hockey and cricket, $15$ play hockey and football, and $12$ play football and cricket. Eight play all the three games. Find the total number of members in the three athletic team.

Answer 1

Let

C= Set of members of cricket team

H= Set of members of Hockey team

F= Set of members of Football team

$n\left(C\right)=21,n\left(H\right)=26,n\left(F\right)=23$

$n(H\cap C)=14,n(H\cap F)=15,n(F\cap C)=12$

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

Total number of members = $n(C\cap H\cap F)$

$=n\left(C\right)+n\left(H\right)+n\left(F\right)-n(C\cap H)-n(H\cap F)-n(C\cap F)+n(C\cap H\cap F)$

$=21+26+29-14-15-12+8=43$