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, a round consists of first randomly partitioning the vertices into c disjoint sets of n vertices each, then adding an edge between every pair of not-yet-joined vertices that lie in the same set. What is the probability that, after y rounds, the result is a complete graph on cn vertices?
I have estimates and solutions to special cases, and it's straightforward to find the probability that a single given student sees all the others, but I don't know how to tackle the question in general. (I do have a very pretty but completely useless expression for the exact answer, which I can supply if there's interest.) In the case c=3, n=25, y=6 it's clear that the answer is "so close to zero that nobody can tell the difference" but I was hoping for a more precise result. Any guidance appreciated.