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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
1
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B
102
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C
38
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D
42
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Solution

The correct option is C 38
Let A,B and C be the set of students opted for Mathematics, Physics and Chemistry respectively then,
A={2,4,6,,140}
n(A)=70
B={3,6,9,,138}
n(B)=46
C={5,10,15,,140}
n(C)=28
(AB)={6,12,18,,138}
n(AB)=23
(AC)={10,20,30,,140}
n(AC)=14
(CB)={15,30,45,,135}
n(AB)=9
(ABC)={30,60,90,120}
n(AC)=4

So,
n(ABC)=
Total students n(ABC)
n(ABC)=n(A)+n(B)+n(C)n(AB)n(BC)n(AC)+n(ABC)
=70+46+2823914+4=102
n(ABC)=140102=38

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