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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:

Example L

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:

Example J

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

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