+275 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.
+2668 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 \(10 \times 9 \times 8 = \color{brown}\boxed{720}\)
+2668 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 \(120 \times (3 \times 2 \times 1) = \color{brown}\boxed{720}\)
