$12$ Students $5$ questions

Question

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?

Answer 1

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.

Answer 2

Answer:

10 Students

Explanation:

Suppose there are 5 questions named as a, b, c, d, e and fifteen students named as 1, 2, 3, ... 12. On the First iteration. 1 gives 3 right answers a, b, c and so correct answered count of each question will become a 1, b 1, c 1, d 0, e 0. The, 2 gives correct answers as d, e, a so now the correct answered count of each question will become a 2, b 1, c 1, d 1, e 1, similarly after five students give answer the correct answered count of each question will become a 3, b 3, c 3, d 3, e 3. Now this iteration will happen again for next five students, and the count will become 6 for each question. So total students that pass the exam are 10 (five in the first iteration and 5 in the second iteration).

Visualization: (a, b, c, d, e are the questions and 1, 2, 3, ... are the students.)