Solve for the number of students who took an exam. An exam consisted of $28$ problems. Each student solved $7$ problems correctly. For every pair of problems solved, there are exactly $2$ students who solved them correctly. How many students took the exam?
I believe I have the correct answer which is $36$ students, but I don't know how to formally set it up to turn in.

Note that each student solves $21$ pairs of problems.
There are $\cfrac {28\times 27}2=14\times 27=21\times 18$ pairs of problems in the set.
If every pair of problems is solved, then every pair is solved twice, giving $21\times 36$ pairs of solutions, which would take $36$ students. Any more than $36$ students and at least one of the problems would have been solved more than twice.
However the problem as stated does not tell us that every pair  of problems has been solved at least once. Two students both solving the same seven problems would also satisfy the conditions as written.
