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problem solving-

Expert replies
by nguyenduong » Sat Aug 10, 2013 10:06 am
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


anyone pls help solution.
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Source: — Problem Solving |

Combinatorics, GMAT Math

by vinay1983 » Wed Aug 28, 2013 8:55 am
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


anyone pls help solution.
I am bumping this for an answer.

I feel it is A.

What is the source?and the OA?
You can, for example never foretell what any one man will do, but you can say with precision what an average number will be up to!
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by ganeshrkamath » Wed Aug 28, 2013 10:35 am
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

anyone pls help solution.
Let the boys be abcdefghi
Total ways of arranging them = 9!
Now abc_def_ghi is the same as abc_def_gih
So taking care of arrangements within a group = 9!/(3!3!3!) = (9*8*7*6*5*4*3!)/(3!3!3!)
= (9*8*7*5*4)/(3!) = (3*4*7*5*4)
= 1680

Also abc_def_ghi is the same as def_ghi_abc
So taking care of arrangements of the groups = 1680/3!
= 1680/6 = 280

Choose E
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by GMATGuruNY » Wed Aug 28, 2013 10:49 am
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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by vinay1983 » Wed Aug 28, 2013 9:27 pm
GMATGuruNY wrote:
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.

I was halfway there. I could realise that it No repetition and order does not matter and I could arrive at 9C3, then I could predict that the teams could be arranged 3! ways, but I failed to apply step 2 and 3, which you have mentioned. Thank Mitch!
You can, for example never foretell what any one man will do, but you can say with precision what an average number will be up to!
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