Let's make it simple and assume every horse has the same probability to win, and let's set it as a 10 horses race.
The event you're interested in is 3 horses in particular arrive in the first three spots in any order and the other 7 horses arrive at the last seven spots.
All the possible ways that the 3 horses can end up in the first three spots is 3! = 3 * 2 * 1 = 6.
All the possible ways that the other 7 horses can end up in the last seven spots is 7! = 5040.
You need those two events to happen, so you simply multiply them: 3! * 7! = 30,240.
Now, the total ways in which all 10 horses can arrive to the finish line is 10! = 3,628,800.
So, the probability of winning that bet is 3! * 7! / 10! = 30,240 / 3,628,800, which is 1 / 120, or 0.00833333333333...
There is one in one hundred and twenty chances of winning that bet.
You know you could have just used the permutations formula, right?
In this case for a trifecta (which btw, it's for the exact order) is simply 10!/ (10 - 3)! and for random order is 10! / 3! × (10 - 3)!.
The chances of getting a trifecta are 1:720 simple as this.
In other words, it is more difficulty with a higher prize if you guess right. Well, that happens in gambling in general, you get a bigger prize if you bet on a roulette number than if you bet on red or black, although in this case it is not exactly like that, but in predicting different outcomes that count as a single bet. There are many types of such bets that you can make in bookmakers, especially in sports betting.
The tiny difference is that the read for guessing 3 numbers in a row is the exact same, no matter what the winning numbers, with horses it's one thing if you guess the right favorites coming home in order and completely different if you guess 3 outsides upsetting any predictions, I guessed 4 times in the last two months a trifecta but never one above 60:1, most in the 20-30 region.
I usually do this kind of betting when I'm sure the two in front are going to dominate but there is nothing interesting about betting on them at 2:1, so you start searching for the third, and with 2-3 pics you might still have a 4-6:1 return even doubling the bets to cover the first two random positions.