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A stable.
paddock
They don't really have a name for a group of race horses as far as I know. Cause generally when racing, they compete against each other. And it's a solo race for the win. So there is no association with the other horses, like for example a team of basketball players. There is no real name for a pack of race horses or a herde of race horses, or horde of race horses..............
Yes, you can place bets until the horses are running.
A Race
In any horse race, bettors are free to bet on as many horses as they would like. Why? Because it does not make sense to bet on too many of the horses, because only one each will win, place, and show. An oversimplified example, but one that illustrates the point: Imagine there are 10 horses in the race, each paying 3-1 to win, 1-1 to place, and 1-3 to show. If you bet on more than 3 horses, you'll lose money since only one of them will win at 3-1 and the others will lose.
720
They can be good race horses against other Morgans, but not against Thoroughbreds or Quarter horses.
Twenty horses can race at a time
This will vary because of several factors. No race is the same. So the purse for the race is not the same. Each place will have its own winnings. The closer to the #1 spot you place in the race, the more the prise money.
If your racing and betting on each others horses its called Racing for Pinks!
The Kentucky Derby has 20 horses. But yeah, it's different from race to race. It's rare that there is more than 20 horses, but it happens. Usually 10-20, but sometimes only 2. But the Grand National has 40 horses.