The number of ways the first three positions of the order of finish can occur in a race with 6 horses and no ties can be calculated using permutation.
Since there are 6 horses competing for the first position, there are 6 possibilities for the first place.
Once the first place is determined, there are 5 remaining horses for the second place, and then 4 remaining horses for the third place.
Therefore, the total number of ways is calculated as follows:
6 (possibilities for first place) × 5 (possibilities for second place) × 4 (possibilities for third place) = 120 ways.
So, there are 120 different ways the first three positions can occur in the race.