Q: Eight athletes participate in a race. Calculate the number of possible outcomes of the race if no ties are allowed.
A: This is the same as asking: how many ways can you put 8 people in a line (considering their order), right? We are just placing people in positions: 1st place, 2nd place, 3rd, …. , all the way down to 8th place… Same thing as lining them up.
So, let us denote our “places” by slots (we have eight of them): _ _ _ _ _ _ _ _.
How many people could go in our first slot? Well, there are 8 people to start with, so we have 8 to pick from:
8 _ _ _ _ _ _ _
How many people could go in our second slot? Well, there are only 7 people left to pick from:
8 7 _ _ _ _ _ _
How many people could go in our third slot? There are only 6 people left, right?
8 7 6 _ _ _ _ _
See the pattern? Eventually we find:
8 7 6 5 4 3 2 1
So, there are 8*7*6*5*4*3*2*1 = 8! ways to arrange 8 different people (8! possible outcomes to the race).