10 teams containing 2 players each compete in a doubles tennis tournament. After the tournament, a gold medal is awarded to the first-place team, a silver medal is awarded to the second-place team, and a bronze medal is awarded to the third-place team. In how many ways could these medals be awarded to 3 of the 10 teams, if there are no ties?

So i started listing out some options.

ABC

ABD

ABE

ABF

.....

This woud take forever to do.

Jasonkiln Aug 7, 2022

#1**0 **

There are 10 ways to choose the first place team

There are 9 ways to choose the second place team

There are 8 ways to choose the third place team.

So, there are \(10 \times 9 \times 8 = \color{brown}\boxed{720}\)

BuilderBoi Aug 7, 2022

#2**0 **

Alternate Solution:

There are \({10 \choose 3} = 120\) ways to choose the 3 teams that win medals.

Of these 3 teams, there are 3 ways to choose the first place team, 2 for the second place team, and 1 for the third place team.

This makes for \(120 \times (3 \times 2 \times 1) = \color{brown}\boxed{720}\)

BuilderBoi Aug 7, 2022