# Grouping classes of objects into more classes?

On page 15 of this Statistics textbook, "Mathematical Statistics and Data Analysis" by John A. Rice, the latter half of Example L has caught my attention:

I've visualized an example of the problem by setting $$m=2$$, so I understand the gist of his explanation. However, I'm not sure what proposition earlier in the chapter is being applied when deriving $$(2m)! \over (m! * 2^m)$$. Then again, when he says, ""...there are $$m!$$ ways to assign the $$m$$ pairs to $$m$$ courts", he is indeed applying Corollary A on page 9.

I feel like I haven't captured the overarching message of this example's latter half, and so it's likely I won't be able to recognize a similar application in the future. If we applied the same logic to Example J on the same page as Example L:

We would we divide $$210$$ by $$3!$$ if we didn't want to commit the subcommittees to specific sizes $${3,2,2}$$?