# in a class of 60 students,23 play hockey,15 play basketball,20 play cricket and 7 play hockey and basketball,5 play cricket and basketball,4 play hockeyand cricket,15 do not play any of the three games. find the number of students who1- play all the three games2-play hockey but not cricket3-play hockey and cricket both,but not basketball

let H, B and C be the set of students who play hockey, basket ball and cricket respectively.

therefore the number of students who play all the three games
$n\left(H\cap B\cap C\right)=n\left(H\cup B\cup C\right)-n\left(H\right)-n\left(B\right)-n\left(C\right)+n\left(H\cap B\right)+n\left(B\cap C\right)+n\left(C\cap H\right)\phantom{\rule{0ex}{0ex}}=45-23-15-20+7+5+4\phantom{\rule{0ex}{0ex}}=45-58+16\phantom{\rule{0ex}{0ex}}=61-58\phantom{\rule{0ex}{0ex}}=3$
2.
the number of students who play hockey but not cricket
$n\left(H\right)-n\left(H\cap C\right)$
= 23 - 4
= 19
3.
play hockey and cricket both , but not basket ball

hope this helps you

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