The pro-am divisions are where the entries really add up quickly.
A very active pro-am student may put in as many as 50 or 75 entries in a single competition. Top Pro-am teachers might have 8 students or more. Contenders for the top teacher prize will usually need a minimum of 300 entries just to get in the running (and some have many more). If you have 5 or 6 teachers competing for that spot, that's at least a couple thousand entries. And don't forget the other 30 or 40 teachers with fewer entries, and it all adds up to several thousand. Now add all the professional and amateur entries, and you can see how 9,400 is a large but perfectly realistic number.
How does a single student get so many entries? Consider the following example:
Jenny is a 35 year-old pro-am student who competes at the silver level in all 4 dance styles (standard, Latin, smooth & rhythm). In each style, she can compete in her level, one level up, and one age division down. In other words, if she's a "full silver lady A2", she can compete in full silver and beginning gold, and in the ladys A1 and A2 age divisions. Every single dance therefore gives her the opportunity to participate in 4 entries.
Standard and Latin each have 5 dances, smooth has 4, and rhythm has about 15 (dependingon the competition). So in the single dance divisions alone, she can enter 29 dances times four, or 116 entries total. And this is just closed single-dance entries. Add open single-dance entries, scholarships, championships, and it's feasible that one student could have as many as 200 pro-am entries.
Of course, no student in his right mind does this, but it illustrates the possibilities. I've heard of top student competitors with over a hundred. It must be a lot of work. 10 years ago I won the top teacher prize at the California Star Ball with a measley 75 entries, and it nearly killed me. I have no idea how the big players with hundreds of entries do it.

Regards,
Jonathan