I think it's mostly just practicing these types of questions.
In your example, we know there are 12 total contestants. We need to pick 1st place, 2nd place, and so on.
How many choices of people do we have for first place? Answer: 12
For second place we have 11 choices
then 10 for 3rd, 9 for 4th and 8 for 5th.
So the overall answer is: (or )
We can also arrive at the answer a different way:
First we choose 5 out of the 12 people (or )
then we rank them: 5 choices for 1st, 4 for 2nd, 3 for 3rd, 2 for 4th, 1 for 5th ( choices total)
So the answer we get is
If you check on your calculator, you will see that these two seemingly different answers are the same.
Important questions to ask yourself:
Does order matter?
Are these objects distinct or indistinct?