Math Help - Permutations & Organized Counting

1. Permutations & Organized Counting

Hello, I would like some help in solving these problems. Please explain how you arrived at your answer if you don't mind.

1. Ten finalists are competing in a race at the Canada games:
a) In how many different orders can the competitors finish the race?
b) How many ways could the gold, silver, and bronze medal be awarded?
c) One of the finalists is a friend from your town. How many of the possible finishes would include them winning a medal?
d) How many possible finishes would leave your friend out of the medal standings?

2. The final score of a soccer game is 6 to 3. How many different scores were possible at half-time?

Thanks for the help!

2. a) There are 10 possible winners.

For each of those 10 winners, there are 9 possible runners who could come 2nd.

For each of those 10 winners and 9 second-placers, there are 8 possible runners who can come 3rd.

And so on.

Thus there are $10 \times 9 \times 8 \times \ldots \times 2 \times 1$ different orders that the race can finish in (if you discard the possibility of a dead heat between two finishers).

This is written $10!$ "10 factorial" and it's about 3.2 million.

b) When you're interested only in gold, silver and bronze you just need the first three, that is $10 \times 9 \times 8$.

That'll do for a start, hope it gives you a clue about how to continue.

3. Hello, NineZeroFive!

2. The final score of a soccer game is 6 to 3.
How many different scores were possible at half-time?

Team A (winning team) could have had any score from 0 to 6: . $\text{7 choices}$

Team B could have had any score from 0 to 3: . $4\text{ choices}$

Therefore, there were: . $7 \times 4 \:=\:28$ possible half-time scores.