Question:
: In a class of 120 students numbered 1 to 120, all even numbered students opt for Physics, whose numbers are divisible by 5 opt for Chemistry and those whose numbers are divisible by 7 opt for Math. How many opt for none of the three subjects
Answer:
In this question, we have to find out the number of students who took at least one of the three subjects
and subtract that number from the overall 120 to get the number of students who did not opt for any of the three subjects.
Number of students who took at least one of the three subjects can be found by finding out A U B U C,
where A is the set of those who took Physics, B the set of those who took Chemistry and C the set of those who opted for Math.
Now, AUBUC = A + B + C - (A n B + B n C + C n A) + (A n B n C)
A is the set of those who opted for Physics = 120/2 = 60 students
B is the set of those who opted for Chemistry = 120/5 = 24
C is the set of those who opted for Math = 120/7 = 17
The 10th, 20th, 30th..... numbered students would have opted for both Physics and Chemistry.
Therefore, A n B = 120/10 = 12
The 14th, 28th, 42nd..... Numbered students would have opted for Physics and Math.
Therefore, C n A = 120/14 = 8
The 35th, 70th.... numbered students would have opted for Chemistry and Math.
Therefore, B n C = 120/35 = 3
And the 70th numbered student would have opted for all three subjects.
Therefore, AUBUC = 60 + 24 + 17 - (12 + 8 + 3) + 1 = 79
Now, the number of students who opted for none of the three subjects = 120 - 79 = 41
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