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Problem solving - 3 groups of 3

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Manager
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Problem solving - 3 groups of 3 [#permalink]  19 Jul 2011, 04:20
00:00

Difficulty:

(N/A)

Question Stats:

20% (00:00) correct 80% (02:26) wrong based on 5 sessions
In how many different ways can a group of 9 people be divided into 3 groups, with each group containing 3 people?

280
1,260
1,680
2,520
3,360

I am looking for an explanation, not the answer.
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Re: Problem solving - 3 groups of 3 [#permalink]  19 Jul 2011, 08:09
from 9 people any 3 can be selected.hence there will be 3 such teams = 9c3/3.

from 6 people any 3 can be selected.hence there will be 2 such teams = 6c3/2

from 3 people any 3 can be selected.hence there will be 1 such team = 3c3/1

thus 9c3 * 6c3 * 3c3/ 3*2*1 = 210.
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Re: Problem solving - 3 groups of 3 [#permalink]  19 Jul 2011, 11:10
Think of it like an experiment:
1. Select 3 people from 9 at random
2. Select another 3 people from the 6 remaining people
3. Select another 3 people for the 3 remaining people

The first thing to notice is that steps 1 and 2 completely determine step 3. The second thing to notice is that since we're selecting teams, order doesn't matter, so we should use combinations
9C3 = (9*8*7)/(3*2*1) =168
6C3 = (6*5*4)/(3*2*1) = 20

By the fundamental counting principle, we should multiply these two numbers to see how many outcomes our experiment has
168*20 = 3,460

BenchPrepGURU
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Re: Problem solving - 3 groups of 3 [#permalink]  19 Jul 2011, 11:11
calculation error!!!!!

168*20 = 3,360

BenchPrepGURU
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Re: Problem solving - 3 groups of 3 [#permalink]  19 Jul 2011, 12:23
Solution:
The first group 9C3=84
The second group 6C3=20
And the last group 3C3=1
In total have 84∙20∙1=1680 ways, but, exist repetitions in ways, then:
( 9C3∙ 6C3∙ 3C3)/3!=1680/6=280
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Re: Problem solving - 3 groups of 3 [#permalink]  22 Jul 2011, 08:28
BenchPrepGURU,

Just wanted to point out some mistakes in your approach so it doesn't mislead someone.

First,
9C3 = (9*8*7)/(3*2*1) = 84 not 168
6C3 = (6*5*4)/(3*2*1) = 20

Second,
1/3rd of the combinations will be repetitions in 3 groups of 3 and 1/2 of the combinations will be repetitions in 2 groups of 3. So you should divided the result be 3 and 2 respectively arriving at 280.
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Re: Problem solving - 3 groups of 3 [#permalink]  22 Jul 2011, 17:45
There are multiple ways to solve this question.

1. Anagram: Let's focus on how to use this method, so that you can reuse it.
Solution:
= 9!/ (3!*3!*3!)
= 1680

2. Direct and more reliable solution is to just visualize one person at a time.
Solution:
= (ways to select 3 out of 9) * (ways to select 3 our of remaining 6) * (ways to select 3 out of remaining 3)
= 9C3*6C3*3C3
= [9!/(6!*3!)] *[6!/(3!*3!)]*1
= 9!/(3!*3!*3!)
= 1680

Ans: "C"

----------------------------------------------------------
I made a mistake early on. I assumed they are 3 distinct group and hence thought order mattered. But this is NOT indicated in the question and MY ASSUMPTION WAS WRONG!

Hence the ans needs to divided by 3! (the number of ways the 3 groups can be rearranged). Hence 1680/3!=280. Ans: A

Thanks manishgeorge for pointing it out.

Hope this helps!

Last edited by abhicoolmax on 22 Jul 2011, 18:58, edited 2 times in total.
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Re: Problem solving - 3 groups of 3 [#permalink]  22 Jul 2011, 18:02
bschool83 wrote:
In how many different ways can a group of 9 people be divided into 3 groups, with each group containing 3 people?

280
1,260
1,680
2,520
3,360

I am looking for an explanation, not the answer.

$$\frac{(3*3)!}{(3!)^3}=\frac{9!}{6^3}=1680$$

Ans: "C"
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Re: Problem solving - 3 groups of 3 [#permalink]  22 Jul 2011, 18:19
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I think the answer is 280 based on the following rationale:

Number of ways for Selecting 3 people out of 9 is 9C3

Number of ways of selecting the next group of 3 is 6C3

Total Number if ways to order three groups is 3! = 6

there for the answer is 9C3*6C3 / 6 = 280

Fluke, have you considered the formula where the order is not important?
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Re: Problem solving - 3 groups of 3 [#permalink]  22 Jul 2011, 18:53
manishgeorge wrote:
I think the answer is 280 based on the following rationale:

Number of ways for Selecting 3 people out of 9 is 9C3

Number of ways of selecting the next group of 3 is 6C3

Total Number if ways to order three groups is 3! = 6

there for the answer is 9C3*6C3 / 6 = 280

Fluke, have you considered the formula where the order is not important?

Good catch:

Should be:
1680/3!=280
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Re: Problem solving - 3 groups of 3   [#permalink] 22 Jul 2011, 18:53
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