Suppose that the 4 racers are Annie, Bob, Carole and Don. I am going to refer to them by the first letter of their names, A, B, C and D.
Suppose you are the score keeper and you are to write down the names in the order that they finish the race. There are 4 possibilities for first place, A, B, C or D. Now, no matter who came first there are 3 possibilities for second place. Thus there are
4 3 = 12
possibilities for first and second place. They are
Again, no matter who came first and second there are two runners left and hence 2 possibilities for third place. Thus there are
4 3 2 = 24
possibilities for first, second and third place.
Finally once the first three runners have finished there is only one possibility for fourth place, and hence there are
4 3 2 1 = 24
possibilities for first, second third and fourth place.
This number 4 3 2 1 is called 4 factorial and written 4!