There are 4 swimmers in a race. there will be a first place, a second place, and third-place prize award. in how many different ways can the 3 prizes be awarded ?

Let's assume there's a fourth place too, and fourth place is the one where it's no prize.

4 3 2 1 (4 3 X X)
4 3 1 2 (^ but 3rd and 4th digit exchanged)
4 2 3 1 (4 2 X X)
4 2 1 3 etc.,.
4 1 2 3
4 1 3 2

As you can see, there are six possibilities assuming first swimmer was fourth.
Now, there would be six possibilities if the first swimmer was third, and same goes for second and first.
Thus, 6 x 4 = 24, there are 24 possibilities to award the 3 prizes (plus 1 no-prize)

Simple Mathematical Way:
4 x 3 x 2 x 1 = 4! = 24
There are 24 ways.