Final answer:
The number of ways a judge can award first, second, and third places in a group of 12 contestants is 1320.
Step-by-step explanation:
In this problem, we need to find the number of ways a judge can award first, second, and third places in a group of 12 contestants.
Since the order matters, we can use the permutation formula:
P(n,r) = n! / (n - r)!
Here, n is the total number of contestants (12) and r is the number of places to be awarded (3).
Plug in the values into the formula:
P(12,3) = 12! / (12 - 3)! = 12! / 9! = 12 × 11 × 10 = 1320
Therefore, there are 1320 ways a judge can award first, second, and third places.