What is the minimum number of students choosing all 4 subjects?

Question

I developed a question for a Mock test by modifying this question.

There are 100 students. 85 chosen Math, 80 chosen Physics, 75 Chosen Chemistry, 70 chosen 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

And I provided the following 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.

I am pretty sure that explanation is correct. We can subtract the complement of these set from universal set, and remaining quantity would be answer because that is common.

But, my friends are arguing that this explanation is not correct as well as question is not correct. I tried to tell them this is correct explanation as my level of knowledge.

Could you please provide an alternative explanation(s) for this question?

One of my friends is arguing that option (D) Data is insufficient is the correct option.

Thanks in advance.

@GaurangTandon, seems relevant but not duplicate. It would be good if someone will give more alternative explanation(s). — Mithlesh Upadhyay, Jan 10 '18 at 09:56
Ok, no worries. The method given by Alexander below is nearly the same as the accepted answer in that question, hence I flagged it as a duplicate. — Gaurang Tandon, Jan 10 '18 at 10:22
@GaurangTandon Indeed, it’s basically the same question, so your flagging is correct. — Alexander Burstein, Jan 10 '18 at 14:36

score 1 · Accepted Answer · answered Jan 10 '18 at 09:17

Sounds correct to me. Let's rephrase it as follows:

There are $100$ students. $15$ students are not taking math, $20$ are not taking physics, $25$ are not taking chemistry, and $30$ are not taking biology. What is the maximum number of students not taking at least one subject?

Obviously, the answer here is $15+20+25+30=90$. So, your answer to your question is correct as well.

What is the minimum number of students choosing all 4 subjects?

1 Answers1