Given the identification of the second-placed horse, the third-placed horse might be any of the other ten horses. So there are 12 × 11 x 10 = 1,320 possible ways for the first, second, and third to occur. This is too many; there must be a way to eliminate some of these possibilities. It's helpful to think about what would happen if one of the twelve horses fell down near the end of the race. There are ten remaining horses, three of which could be identified as first, second, and third place. The rest are all interchangeable, so there are still only ten possibilities. This means that there are at least ten ways for someone to win with twelve horses entered in the race. There are several ways to reduce the number of possibilities further. For example, we know that no two horses can finish in first place, so there can't be more than two of them. The same goes for the third place horse. There can't be more than two first place horses or more than two third place horses. This leaves us with at most four different ways they can finish without putting more than two horses in first or third place. Another way to look at this is to note that since there are twelve horses entering the race, each place will be filled by one of them.