Question

In a higher secondary class, 66 plays football, 56 plays hockey, 63 plays cricket, 27 play both football and hockey, 25 plays hockey and cricket, 23 plays cricket and football and 5 do not play any game. if the strength of class is 130. Calculate
(i) the number who play only two games
(ii) the number who play only football
(iii) number of student who play all the three games  

Answer 1

Answer:

(i)The number of students who play only two games is 30

(ii)The number of students who play only foot ball is 31

(iii)The number of students who play only all the 3 games is 15

Step-by-step explanation:

Let F, H and C be the set of students who play foot ball, hockey and cricket respectively.

Let x be the number of students who paly all the 3 games

since 5 students don't paly any game,

the number of students of who play atleast any one of the three games is 130-5=125

$\bf\,n(F\cup\,H\cup\,C)=n(F)+n(H)+n(C)-n(F\cap\,H)-n(H\cap\,C)-n(F\cap\,C)+n(F\cap\,H\cap\,C)$

$\implies\,125=66+56+63-27-25-23+x$

$\implies\,125=185-27+x-25+x-23+x+x$

$\implies\,125=185-75+x$

$\implies\,125=110+x$

$\implies\,\bf\,x=15$

(i)The number of students who play only two games

=(27-x)+(25-x)+(23-x)

=75-3x

=75-45

=30

(ii)The number of students who play only foot ball

=66-(27-x+x+23-x)

=66-(27+23-x)

=66-(27+23-15)

=66-35

=31

(iii)The number of students who play only all the 3 games

=15