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m24 q16

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m24 q16 [#permalink] New post 16 Apr 2010, 18:26
Hi All,

I couldnt find this problem in the forum...so if I missed it...I apologize!

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
[Reveal] Spoiler:
10


I am not quite sure about the answer...shouldnt it be as simple as 6C3...do we really have to consider leaving groups behind etc...I have never encountered this kind before...any further explanation would help.
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Re: m24 q16 [#permalink] New post 16 Apr 2010, 21:27
Should be 6C3 / 2

Because half of the combination would be repeated in the other group.
For example, if group 1 has "A B C", group 2 must be "D E F"
Hence, the combination "D E F" in group 1 should be double counted.
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Re: m24 q16 [#permalink] New post 22 Oct 2010, 04:26
calvinhobbes wrote:
Should be 6C3 / 2

Because half of the combination would be repeated in the other group.
For example, if group 1 has "A B C", group 2 must be "D E F"
Hence, the combination "D E F" in group 1 should be double counted.



6C3 simply means selecting 3 out of 6 and the order is not important.
Why divide the original expression by 2?

Say the set is: {A,B,C,D,E,F}
Using 6C3 select {A,B,D} and form 1 group - the other group would be formed automatically{C,E,F} - I dont see any "doubling".
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Re: m24 q16 [#permalink] New post 22 Oct 2010, 06:18
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vicksikand wrote:
calvinhobbes wrote:
Should be 6C3 / 2

Because half of the combination would be repeated in the other group.
For example, if group 1 has "A B C", group 2 must be "D E F"
Hence, the combination "D E F" in group 1 should be double counted.



6C3 simply means selecting 3 out of 6 and the order is not important.
Why divide the original expression by 2?

Say the set is: {A,B,C,D,E,F}
Using 6C3 select {A,B,D} and form 1 group - the other group would be formed automatically{C,E,F} - I dont see any "doubling".


Not so.

For example if we choose with C^3_6 the group {ABC} then the group {DEF} is left and we would 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!.

Check similar questions:
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: m24 q16   [#permalink] 22 Oct 2010, 06:18
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