A cohort in a school consists of 75 students who study for 6 years. Each year, the students are randomly distributed into 3 classrooms of 25 students each. What is the probability that, after 6 years, each student has at some point been in a classroom with every other student?

More generally: Starting with an edgeless (undirected) graph on $cn$ vertices, let a **round** consist of first randomly partitioning the vertices into $c$ disjoint sets of $n$ vertices each, then adding an edge between every pair of nonadjacent vertices that lie in the same set. What is the probability that, after $y$ rounds, the result is a complete graph?

I asked this question on math.stackexchange but received no fully useful response (see here, where I've also posted an answer with further discussion and generalization and partial "solutions"). I'd especially like to know about tools for the exact answer, but approximations or bounds would also be interesting.

The particular case above was posed by a friend, who teaches in a school with those values of $c$, $n$, and $y$. In that particular case the answer is easily seen to be "Don't hold your breath, pal."