Kendo match schedules fair?

So once in a while I go to a kendo match in a delegation of our dojo. Today, I went to the Ijima-cup tournament; named after Ijima sensei, who has meant a great deal for the development of kendo in the Netherlands. Our dojo sent 8 members to the competition.

There were no less than 29 groups in the first stage of the tournament. All but one group contained three kendoka. To our surprise two of our dojo members were in group number 20. All the rest were divided over different groups, so the rest of us didn't have to fight friends from the dojo.
The reason why it surprised us was that until today, we were much less fortunate. Some began even to suspect that there was some evil scheming at work. I told them that it was probably just a matter of probability. However, I have not yet come to make a model.
So this will be my next 'challenge':

What is the probability distribution of X, the number of groups with more than one fellow dojo member, when m members of our dojo are randomly assigned to any one of g groups, each consisting of n group competitors?

At the moment, I am tired. I still have some preparations to do for tomorrow, so I will not try to figure out the answer just now.

P.S. The tournament went well. I stranded in third round. After a period in which my kendo seemed to get less flexible, it slowed down, cramped up and only felt frustrating, today I did get "in the flow" again and I could do some fluid, strong and fast kendo. Most of our dojo members did well.