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# Counting QUestion

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119
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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
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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
+2532
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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
+1

Thank you!

Aug 8, 2022