Question

In a class of $140$ students numbered $1$ to $140$, all even numbered students opted mathematics course, those whose number is divisible by $3$ opted Physics course and those whose number is divisible by $5$ opted Chemistry course. Then the number of students who did not opt for any of the three courses is?

• A. $102$
• B. $42$
• C. $1$
• D. $38$

Answer 1

The correct option is D $38$

Let $n\left(A\right)=$ number of students opted Mathematics $=70$,
$n\left(B\right)=$ number of students opted Physics $=46$,
$n\left(C\right)=$ number of students opted Chemistry $=28$,
$n(A\cap B)=23$,
$n(B\cap C)=9$,
$n(A\cap C)=14$,
$n(A\cap B\cap C)=4$,
Now $n(A\cup B\cup C)$
$=n\left(A\right)+n\left(B\right)+n\left(C\right)-n(A\cap B)-n(B\cap C)-n(A\cap C)+n(A\cap B\cap C)$
$=70+46+28-23-9-14+4=102$
Si number of students not opted for any course $=$ Total $-n(A\cap B\cap C)$
$=140-102=38$.