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# How many ways are there to split a group of 6 boys into two

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How many ways are there to split a group of 6 boys into two [#permalink]

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25 Nov 2010, 17:06
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Question Stats:

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How many ways are there to split a group of 6 boys into two groups of 3 boys each? (The order of the groups does not matter)

A. 8
B. 10
C. 16
D. 20
E. 24
[Reveal] Spoiler: OA

Last edited by Bunuel on 09 Jul 2013, 10:15, edited 1 time in total.
Edited the question.
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Re: ways to split a group of 6 boys into two groups of 3 boys ea [#permalink]

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25 Nov 2010, 17:13
Expert's post
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yufenshi wrote:
How many ways are there to split a group of 6 boys into two groups of 3 boys each? (The order of the groups does not matter)

8
10
16
20
24

The official answer is B, but I don't understand why in this case we need to divide by 2. It seems to me to be the same questions as: how many ways to choose 3 people out of 6 people. in that case it would be 20.

Thanks!

GENERAL RULE:
1. The number of ways in which $$mn$$ different items can be divided equally into $$m$$ groups, each containing $$n$$ objects and the order of the groups is important is $$\frac{(mn)!}{(n!)^m}$$

2. The number of ways in which $$mn$$ different items can be divided equally into $$m$$ groups, each containing $$n$$ objects and the order of the groups is NOT important is $$\frac{(mn)!}{(n!)^m*m!}$$.

BACK TO THE ORIGINAL QUESTION:
In original question as the order is NOT important, we should use second formula, $$mn=6$$, $$m=2$$ groups $$n=3$$ objects (people):
$$\frac{(mn)!}{(n!)^m*m!}=\frac{6!}{(3!)^2*2!}=10$$.

This can be done in another way as well: $$\frac{C^3_6*C^3_3}{2!}=10$$, we are dividing by $$2!$$ as there are 2 groups and order doesn't matter.

For example if we choose with $$C^3_6$$ the group {ABC} then the group {DEF} is left and we have two groups {ABC} and {DEF} but then we could choose also {DEF}, so in this case second group would be {ABC}, so we would have the same two groups: {ABC} and {DEF}. So to get rid of such duplications we should divide $$C^3_6*C^3_3$$ by factorial of number of groups - 2!.

This concept is also discussed at:
combinations-problems-95344.html?hilit=dividing%20objects%20order#p734396
split-the-group-101813.html?hilit=split
9-people-and-combinatorics-101722.html?hilit=divided%20equally%20into#p788744
ways-to-divide-99053.html?hilit=divided%20equally%20into#p763471

Hope it helps.
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Re: ways to split a group of 6 boys into two groups of 3 boys ea [#permalink]

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25 Nov 2010, 21:49
and after going through Bunuel's explanation, to ensure that you have understood the concept, try the question in this post: http://gmatclub.com/forum/combination-105384.html

It the same reason why you do not multiply by 2 in the question in this post.
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Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews Senior Manager Joined: 08 Nov 2010 Posts: 417 WE 1: Business Development Followers: 7 Kudos [?]: 97 [0], given: 161 Re: ways to split a group of 6 boys into two groups of 3 boys ea [#permalink] ### Show Tags 27 Nov 2010, 13:24 if i used the 6!/3!*(6-3)! * 1/2 is that ok as well? _________________ Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 6970 Location: Pune, India Followers: 2028 Kudos [?]: 12753 [0], given: 221 Re: ways to split a group of 6 boys into two groups of 3 boys ea [#permalink] ### Show Tags 27 Nov 2010, 13:44 144144 wrote: if i used the 6!/3!*(6-3)! * 1/2 is that ok as well? Absolutely! You are doing 6C3 * (1/2). _________________ Karishma Veritas Prep | GMAT Instructor My Blog Get started with Veritas Prep GMAT On Demand for$199

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Re: ways to split a group of 6 boys into two groups of 3 boys ea [#permalink]

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27 Nov 2010, 14:12
VeritasPrepKarishma wrote:
144144 wrote:
if i used the
6!/3!*(6-3)! * 1/2 is that ok as well?

Absolutely! You are doing 6C3 * (1/2).

I'd say it depends on the logic behind this formula.
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Re: How many ways are there to split a group of 6 boys into two [#permalink]

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18 Aug 2015, 06:33
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Re: How many ways are there to split a group of 6 boys into two [#permalink]

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26 Nov 2015, 16:40
Or simply put it in an anagram grid as: BBB|BBB , so we have 6!/3!*3!*2!=6*5*4/3*2*2=10.
Re: How many ways are there to split a group of 6 boys into two   [#permalink] 26 Nov 2015, 16:40
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