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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.

 Aug 7, 2022
 #1
avatar+2437 
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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 \(10 \times 9 \times 8 = \color{brown}\boxed{720}\)

 Aug 7, 2022
 #2
avatar+2437 
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 \(120 \times (3 \times 2 \times 1) = \color{brown}\boxed{720}\)

 Aug 7, 2022
 #3
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+1

Thank you!

 Aug 8, 2022

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