Assume a standardized test has $15$ multiple choice questions, with $3$ options each. A school has some students and wishes for at least one of the students to get $10$ correct answers. Since no one in the school knows any math they are going to cheat and tell each student which answers he has to give without knowing what the actual correct answers are.
The problem is to find a small number of students $n$ such that it is possible to do this and guarantee at least one of the students gets at least $10$ correct.
For example one way to do it is with $n=3\cdot2^{15}-1$ students, where we do all possible exams with at most two different options, since one of the options appears at most \$5\$ times we can get at least one test with $10$ correct solutions.
Another possible $n$ is $a^5$ where we can get a set of $a$ tuples of length $3$ such that for every option of length $3$ there is a tuple in the set that matches in at least two positions (so basically a solution to the problem with length $3$ and at least $2$ correct). It seems that the vectors $(1,1,1),(2,2,2),(3,3,3)$ along with $(0,1,1),(1,2,2),(2,0,0)$ is guaranteed to get at least two correct for each sequence of three answers, so $6^5$ is also a valid $n$.