Final answer:
There are 31 x 30 x 29 = 26,970 ways the first, second, and third place ribbons can be awarded.
Step-by-step explanation:
In this problem, we need to determine the number of ways the first, second, and third place ribbons can be awarded to the 31 horses competing in a show.
For the first place ribbon, there are 31 possible horses that can win. After choosing the first place horse, there are 30 remaining horses that can win the second place ribbon. Finally, there are 29 horses left to win the third place ribbon.
Therefore, using the multiplication principle, the total number of ways the ribbons can be awarded is 31 x 30 x 29 = 26,970 ways.