P vs NP problem (Student example)
Formally, this is a special case of the "$k$-independent set problem", and can be stated as follows:


  Given a graph with 400 vertices, can you pick 100 vertices such that no two of them are adjacent?


(A "graph" is a set of vertices (students) and edges (students who are incompatible). Two vertices are "adjacent" if there is an edge between them (they're incompatible).)

You can find easy solutions for some groups of students: for instance, if there are no incompatibilities at all, any choice of 100 students will work. But the difficult problem is solving this in general, coming up with an algorithm that can solve it for any selection of $k$ students from a group of $n$, with a "polynomial runtime" (which basically means it can be solved reasonably quickly even for large values of $n$).

"Proving" that the problem is solved involves a formal mathematical proof, a way of demonstrating that your algorithm works for any group of students, no matter what the incompatibilities are, no matter how many there are, and takes only polynomial time to run.

Examples won't work as a proof, because no matter how many examples you show, it's always possible that your algorithm would fail on the next one you haven't tried.

If you're actually interested in approaching this problem, the formal problem description explains it more thoroughly. (You will need some mathematical background though.)