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A tennis coach divides her 9-player squad into three 3-player groups with each player in only one group. How many different sets of three groups can be made?

May 2, 2020

#1
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For the first team, the coach has $$9$$ possible choices and picks $$3$$ players. This is the same as $$\dbinom{9}{3}$$ as order doesn't matter in a group. Thus, there are $$\dbinom{9}{3} = 84$$ ways to pick the first team.

For the second team, the coach has $$6$$ possible choices left and again picks $$3$$ players. This is $$\dbinom{6}{3}$$, due to the reasoning above. Thus, there are $$\dbinom{6}{3} = 20$$ ways to pick the second team.

For the third team, the coach only has $$3$$ choices for players, and there is only way to choose 3 people from 3 people. Thus, there is $$\dbinom{3}{3} = 1$$ to pick the third and final team.

Multiplying it all together, we get $$84 \cdot 20 \cdot 1 = \fbox{1680}$$ ways! :D

May 2, 2020
#2
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Great explanation CentLord!

LuckyDucky  May 2, 2020
#3
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:D Thanks!

CentsLord  May 3, 2020