Probability involving unique group combinations
If I have $30$ objects and $5$ buckets that each hold $6$ objects, how many times could I put the objects into the buckets without an object being in the bucket with an object it has previously been grouped with? So, for each round you would empty all of the objects from the buckets and place them into buckets again (they could be in the same bucket multiple times, just not see another object multiple times).
Edit: A better example would be $120$ objects in $20$ buckets that hold $6$ objects. No object can ever be in a bucket with another object that it has seen previously.