Question

In a class of $80$ students numbered $1$ to $80$, all odd numbered students opt for cricket, students whose numbers are divisible by $5$ opt for football and students whose numbers are divisible by $7$ opt for hockey. The number of students who do not opt any of the three games, is -

• A. $13$
• B. $24$
• C. $28$
• D. $52$

Answer 1

The correct option is C $28$

Set of students who opt Cricket = A
Set of students who opt Football = B
Set of students who opt Hockey = C

From the given information,
$n\left(A\right)=40,n\left(B\right)=16,n\left(C\right)=11$
$n(A\cap B)=8,n(A\cap C)=6,n(B\cap C)=2$
$n(A\cap B\cap C)=1$

Using $n(A\cup B\cup C)=n\left(A\right)+n\left(B\right)+n\left(C\right)-n(A\cap B)-n(B\cap C)-n(C\cap A)+n(A\cap B\cap C)$
$=40+16+11-8-6-2+1=52$

And the number of students who do not opt any of the three games are given by $n(A^c\cap B^c\cap C^c)=n\left(U\right)-n(A\cup B\cup C)=80-52=28$