nguyenduong wrote:Nine highschool boys gather at the gym for a game of mini-volleyball. Three teams of 3 people
each will be created. How many ways are there to create these 3 teams?
(A)
27
(B)
51
(C)
90
(D)
175
(E)
280
Approach 1:
Team A:
From the 9 boys, the number of ways to choose 3 = 9C3 = (9*8*7)/(3*2*1) = 84.
Team B:
From the remaining 6 boys, the number of ways to choose 3 = 6C3 = (6*5*4)/(3*2*1) = 20.
Team C:
From the remaining 3 boys, the number of ways to choose 3 = 3C3 = (3*2*1)/(3*2*1) = 1.
To combine the options for each team, we multiply:
84*20*1.
Since the ORDER of the teams doesn't matter -- ABC is the same 3 teams as BCA-- we divide by the number of ways the 3 teams can be ARRANGED (3!):
(84*20*1)/(3*2*1) = 280.
The correct answer is
E.
Approach 2:
The first boy selected must be combined with a pair of boys formed from the other 8 boys.
From the other 8 boys, the number of ways to choose 2 = 8C2 = (8*7)/(2*1) = 28.
6 boys remain.
The next boy selected must be combined with a pair of boys formed from the other 5 boys.
From the other 5 boys, the number of ways to choose 2 = 5C2 = (5*4)/(2*1) = 10.
3 boys left.
The next boy selected must be combined with a pair of boys formed from the other 2 boys.
From the other 2 boys, the number of ways to choose 2 = 2C2 = (2*1)/(2*1) = 1.
To combine the options for each boy selected, we multiply:
28*10*1 = 280.
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