Question

Among $43$ members of three athletic teams in a school, $21$ are in cricket team, $26$ are in hockey team and $29$ are in football team. Among them, $14$ play hockey and cricket, $15$ play hockey and football and $12$ play football and cricket. If each member plays at least one game, then the number of members who play all the three games, is

• A. $12$
• B. $8$
• C. $6$
• D. None of these

Answer 1

The correct option is B $8$

Let $C,H$ and $F$ represent the players of cricket, hockey and football respectively.
Given $n(C\cup H\cup F)=43,$
$n\left(C\right)=21,n\left(H\right)=26,n\left(F\right)=29,n(C\cap H)=14,n(H\cap F)=15,n(C\cap F)=12$

$n(C\cup H\cup F)=n\left(C\right)+n\left(H\right)+n\left(F\right)-\left[n\right(C\cap H)+n(H\cap F)+n(C\cap F\left)\right]+n(C\cap H\cap F)\Rightarrow 43=21+26+29-(14+15+12)+n(C\cap H\cap F)\therefore n(C\cap H\cap F)=8$

Hence, the number of players playing all the three games is $8.$