question_answer

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

Correct Answer: D

Solution :

\[n(P)=\left[ \frac{140}{3} \right]=46\]

\[n(C)=\left[ \frac{140}{5} \right]=28\]

\[n(M)=\left[ \frac{140}{2} \right]=70\]

\[n(P\cup C\cup M)\]

\[=n(P)+n(C)+n(M)-n(P\cap C)-n(C\cap M)\]

\[-n(M\cap P)+n(P\cap M\cap C)\]

\[=46+28+70-\left[ \frac{40}{15} \right]-\left[ \frac{140}{10} \right]-\left[ \frac{140}{6} \right]+\left[ \frac{140}{30} \right]\]

\[=144-9-14-23+4=102\]

So, required number of students \[=140-102=38.\]