# $12$ Students $5$ questions

In a class there are $12$ students and their instructor asked $5$ questions for an exam. In order to pass this exam, every student needs to answer at least $3$ questions correctly.

It is known that every question is answered correctly by $6$ students;

At most how many students might have passed this exam?

There are 5 questions, each of which is answered correctly by 6 students. That makes 30 correct answers in total.

With three correct answers required in order to pass, this means that the maximum number of passing students is (30/3 ==) 10.

Edit, with an example table of which students get which question correct:

Each row is one of the 12 students. Each column is one of the five questions. The X marks which questions each student got right.

   1   2   3   4   5
1  X   X   X
2  X   X       X
3  X       X       X
4  X   X       X
5  X       X   X
6  X           X   X
7      X   X       X
8      X       X   X
9      X   X       X
10         X   X   X
11
12


Note that each of the five questions is answered correctly six times (has six Xs in its column). Students 1-10 each answer three questions correctly (three Xs in their row), and therefore pass. Students 11-12 answer zero questions correctly, and therefore fail.

Thus, ten students pass, two fail.

This mapping of students to the answers they got correct obviously isn't the only way to make a mapping with ten students passing; it's pretty trivial to just naively fill in this sort of table in a bunch of different ways, and then move the correct answers around to make it obey the "three correct answers per students, six students per answer" constraints. I threw this one together in about thirty seconds; a lot more time went into formatting it for the post, than into constructing it.

I found that filling out this table feels a lot like solving a vastly simplified Eight Queens problem. When I was asked to provide the table, I assumed that I had missed some tricky gotcha of the question in my original, maths-based answer, and that I was going to have to rethink my whole approach after discovering what that gotcha was.. but.. nope! Doesn't seem to be any gotcha, here.

• And the minimum number that can pass... obviously 11 can get 2 each and fail, the last gets 8 and aces it. OK that's not possible... 10 can get 2 each, last two get 5 each. So just two pass. Jan 1, 2018 at 10:32
• I find it interesting that the number of students who took the test actually doesn't matter for calculating the answer to the puzzle. Based upon the "six correct answers per question", "five questions", and "three correct answers to pass" clauses, only a maximum of 10 students can possibly have passed the test, whether there were 12 or 12,000,000 students taking the test. Jan 1, 2018 at 10:33
• could you share a table showing that it is actually 10?
– Oray
Jan 1, 2018 at 10:39
• About the last paragraph: the original answer was incomplete because you only showed that it couldn't be more than 10 students, but you didn't show that it could be 10 students. With the table it's now a complete solution.
– ffao
Jan 1, 2018 at 13:59
• To prove that a number x = 10, you have to prove that it is less than 11 (as you did), but then it can still be 9,8 or 7. You also have to show that it is greater than 9 (as the table does). Any mathematician would tell you the same. This is not a matter of personal opinion.
– ffao
Jan 2, 2018 at 1:51