Question

In a group of children $35$ play football out of which $20$ play football only, $22$ play hockey; $25$ play cricket out of which $11$ play cricket only. Out of these $7$ play cricket and football but not hockey, $3$ play football and hockey but not cricket and $12$ play football and cricket both. How many play all three games?

• A. $5$
• B. $2$
• C. $12$
• D. $60$

Answer 1

The correct option is A $5$

Let $F,H$ and $C$ denote the no. of children who play Football, Hockey and Cricket respectively.
Given $n\left(F\right)=35,n\left(H\right)=22,n\left(C\right)=25$ and $n(F\cap C\cap H^{\prime })=7,n(F\cap H\cap C^{\prime })=3,n(F\cap C)=12$
but $n(F\cap C\cap H^{\prime })=n(F\cap C)-n(F\cap C\cap H)$
$\Rightarrow 7=12-n(F\cap C\cap H)$
$\Rightarrow n(F\cap C\cap H)=5$
$\therefore$ No of children who play all three games is $5$
Hence, option A