Question

Among a group of students, $50$ played cricket, $50$ played hockey and $40$ played volley ball. $15$ played both cricket and hockey, $20$ played both hockey and volley ball, $15$ played cricket and volley ball and $10$ played all three. If every student played at least one game, then

• A. The sum of the number of students who played only cricket, only hockey and only volley ball is $70$
• B. The sum of the number of students who played only cricket, only hockey and only volley ball is $75$
• C. The number of students who played only cricket is $30$
• D. The number of students who played only volley ball is $15$

Answer 1

Answer: (A), (C), (D)

The number of students who played only volley ball is $15$

Set $C,H$ and $V$ represents number of players playing cricket, hockey and volleyball respectively.

From the above venn diagram:
$n\left(C\right)=50\Rightarrow a+d+g+e=50n\left(H\right)=50\Rightarrow c+d+g+f=50n\left(V\right)=b+e+g+f=40n(C\cap H)=15\Rightarrow d+g=15n(H\cap V)=20\Rightarrow g+f=20n(C\cap V)=15\Rightarrow e+g=15n(C\cap V\cap H)=10\Rightarrow g=10$
From avove expressions:
$e=5,f=10,d=5,a=30,b=15,c=25$
Thus, Sum of the number of players who only play cricket, only play hockey and only play volleyball
$=a+b+c=30+15+25=70$
Also, number of players playing only cricket $=30$
And, number of players playing only Volley ball $=15.$