Suppose there is 1 person in the race. There is only one possible combination.
A 2nd racer ( #2) is in the race. This person can be placed in 2 places(in front of 1 , or behind 1) There are now 2 possible combinations 12 21
a 3rd person enters the race. Racers 1 and 2 are in place, number 3 has 3 possible places to fit in(in the front, between 1 and 2, or in the back. Because there are 2 possible positions for 1 and 2, there are 3*2 possible combinations for 3 racers
12 becomes 312, 132, or 123
21 becomes 321, 231, or 213
a 4th racer enters, there are 4 places this new racer can fit in There will be 4*6 possible combinations
n racers have n! possible positions