Logic Problem: $1000$ students took part in a mathematical competition which had only $4$ questions

Question

I have encountered this word problem recently, which has confused me a bit. The problem reads:

"$1000$ students took part in a mathematical competition which had only $4$ questions. $900$ answered correctly to the first question, $800$ answered correctly to the second question, $700$ answered correctly to the third question, and $600$ answered correctly to the fourth question. None of the participants answered correctly to all four of the questions. The students who solved the third and fourth questions were awarded a medal. How many students got a medal:

a) 700, b) 500, c) 300, d) 650, e) 200.

My approach

$300$ students answered wrong to the third question and $400$ students answered wrong to the fourth. $300+400=700$ and $1000-700=300$, which I know is the correct answer (I have the answers). However, I think that this is the maximum number of students that can get an award.

How can we show that 200 cannot be the answer?

Answer 1

You know exactly 1000 wrong answers were given, and each of 1000 candidates gave at least one incorrect answer. Nobody can have given two wrong answers, or we would have had at least 1001 wrong answers. Therefore everyone gave exactly one wrong answer snd got all the others right.

Of 1000, 300 got answer 3 wrong and all others right, 400 got answer 4 wrong and all others right, 300 got either 1 or 2 wrong and got 3 and 4 right. 300 prizes.

Answer 2

I disagree with your reasoning: you are assuming that the $300$ students who answered wrong to the third question and the $400$ students who answered wrong to the fourth question are totally separate from each other, but there's no guarantee of that. Ignoring for a moment the information about the first and second question, it seems possible that all the students who answered the third question wrong also answered the fourth question wrong, in which case there would be $600$ medals awarded. I believe $300$ is actually the minimum number of students that can get an award.

If that's true, then you need to show that more than $300$ students cannot have won a medal. You haven't used any information about the first and second questions yet; maybe that will be helpful in finishing things off.