There are 7 possibilities for first place. For each of those, there are six remaining possibilities for second place. And for each first/second place combination, there are five remaining possibilities for third place. So, there are a total of 7 * 6 * 5 = 210 possible finishes.
This can also be computed using combinatorics, namely the permutations of 7 things, 3 at a time, with no repeats allowed, which is written as P(7,3) or 7P3, and is computed as follows:
P(n,r) = n! / (n-r)!
P(7,3) = 7! / (7-3)! = 7! / 4! = 5040 / 24 = 210
