7 people enter a race. There are 4 types of medals given as prizes for completing the race. The winner gets a platinum medal, the runner-up gets a gold medal, the next two racers each get a silver medal, and the last 3 racers all get bronze medals.
What is the number of different ways the medals can be awarded?

a) 7!/3! b) 7!/2!3! c) 112!3! d) 762!3! e) 765*4

The number of ways of awarding the medals is equal to the number of ways of choosing a winner, a runner up, the next two racers and the last 3 racers from our group of 7 people (the different positions of the racers).

To count this, we use the multiplication principle from combinatorics. It says that if there are A ways of doing something and B ways of doing another thing, there are A×B ways of doing both things.

First, there are 7 people and we have to choose 1 winner. This can be done in 7 ways. This person earns the platinum medal

After choosing the winner, 6 people remain. We have to choose 1 runner-up from this group, which can be done in 6 ways.

Now, 5 people remain. We have to choose 2 of them for the silver medal. Since the order between them doesn't matter (they get a silver medal regardless who was 3rd or 4th) the number of ways of doing this is equal to the binomial coefficient [tex]\binom{5}{2}=\frac{5!}{2!3!}[/tex].

3 people remain. They will be the last racers, so there is only 1 way of choosing them.

Using the multiplication principle, the number of ways of awarding the medals is [tex]7(6)\frac{5!}{2!3!}=\frac{7(6)(5!)}{2!3!}=\frac{7!}{2!3!}[/tex]