If 4 people are in a race, how many different placements, i.e., 1st, 2nd, 3rd, 4th, can there be and what is the equation? Tammy Elementary Parent Hi Tammy, 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 AB AC AD BA BC BD CA CB CD DA AB DC 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! Penny