GATE | GATE CS Mock 2018 | Question 10

There are 100 students. 85 choose Math, 80 choose Physics, 75 choose Chemistry, 70 choose Biology. What is the minimum number of students choosing all 4 subjects? Assume each student chosen at least one subject.

(A) 15
(B) 10
(C) 5
(D) Data is insufficient


Answer: (B)

Explanation: Let A,B,C,D ⊂ {1,2,…,100} be the four sets, with |A| = 85, |B| = 80, |C| = 75, |D| = 70. Then we want the minimum size of (A∩B∩C∩D). Combining the fact that
|A∩B∩C∩D| = 100 − |A’∪B’∪C’∪D’|
where A’ refers to A complement, along with the fact that for any sets |X∪Y|≤|Y|+|X| we see that
|A∩B∩C∩D| ≥ 100 − |A’|−|B’|−|C’|−|D’| = 10.

So, option (B) is correct.l

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