# Challenging Counting problem

I have a counting problem for work that we've been trying to solve for a bit now, and have finally come here for help. Here is the problem:

Given 100 people, split the 100 people into groups of 4. Now we have 25 groups of 4 people. The question is, how many times can the 100 people be split into 25 groups of 4 so that no one is in a group with the same person twice?

Ideally, we are looking for a general solution to the problem. We have worked out a couple of smaller versions by hand. For instance, for 16 people split into groups of 4, the groups can be made 3 three times.

Any ideas?

• do you agree that 50 groups of 2 is the same as permutations without a fixed point ? Line up the first 50 people. Match them with the second. Call that arrangement of the second 1,...,50. Now you want to permute the first so that 1 is never in the 1's place etc. I think that with more groups you may have latin rectanges. I have seen asymptototics for same (C. Stein) but never the exact version. – mike Apr 24 '12 at 18:51

The names of what actually needs to be looked for are somewhat scattered. For example, 36 golfers in foursomes can play for 11 days. This is a 4-RGDD of type $2^{18}$.