In a class of 120 students numbered 1 to 120, there are 3 extracurricular activities to choose from. All even numbered students opt for Dance, whose numbers are divisible by 5 opt for Music and those whose numbers are divisible by 7 opt for Debate. How many opt for none of the three subjects?

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Solution

The correct option is B 41
option (b)

A is the set of those who opted for Dance = $\frac{120}{2} = 60$ students.

B is the set of those who opted for Music = $\frac{120}{5} = 24$ .

C is the set of those who opted for Debate = $\frac{120}{7} = 17$ .

The 10th, 20th, 30th…… numbered students would have opted for both Dance and Music. $\frac{120}{10} = 12$ .

The 14th, 28th, 42nd…. Numbered students would have opted for Dance and Debate.= $\frac{120}{14} = 8$ .

The 35th, 70th …. Numbered students would have opted for Music and Debate = $\frac{120}{35} = 3$ .

And the 70th numbered student would have opted for all three subjects.

S=101, II+3III = 23

III= 1, S-X= 20+2

101-22 = X

X=79

Therefore, number of students who did not opt for any of the three activities = 41

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