Final answer:
Using the permutation formula, there are 720 different ways for 10 horses to finish in the top three places in a race.
Step-by-step explanation:
The question involves calculating the number of different possible ways 10 horses can finish first, second, and third in a race. This type of problem is best approached using permutations, specifically a permutation without repetition, since we are considering the order of the horses and a horse cannot place more than once.
To calculate the total permutations of 10 horses taking the top three places, we apply the formula for permutations of n items taken r at a time, which is P(n,r) = n! / (n-r)!. In this case, n is the total number of horses (10), and r is the number of places (3).
P(10,3) = 10! / (10-3)! = 10! / 7! = (10 \u00d7 9 \u00d7 8) = 720.
Therefore, there are 720 different ways for 10 horses to come in first, second, or third.