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 who

1- play all the three games

2-play hockey but not cricket

3-play hockey and cricket both,but not basketball

$n\left(H\right)=23;n\left(B\right)=15;n\left(C\right)=20\phantom{\rule{0ex}{0ex}}n(H\cap B)=7;n(C\cap B)=5;n(H\cap C)=4\phantom{\rule{0ex}{0ex}}n(H\cup B\cup C)=60-15=45$

therefore the number of students who play all the three games

$n(H\cap B\cap C)=n(H\cup B\cup C)-n\left(H\right)-n\left(B\right)-n\left(C\right)+n(H\cap B)+n(B\cap C)+n(C\cap H)\phantom{\rule{0ex}{0ex}}=45-23-15-20+7+5+4\phantom{\rule{0ex}{0ex}}=45-58+16\phantom{\rule{0ex}{0ex}}=61-58\phantom{\rule{0ex}{0ex}}=3$

2.

the number of students who play hockey but not cricket

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

= 23 - 4

= 19

3.

play hockey and cricket both , but not basket ball

$=n(H\cap C)-n(H\cap C\cap B)\phantom{\rule{0ex}{0ex}}=4-3\phantom{\rule{0ex}{0ex}}=1$

hope this helps you

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