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# In how many ways can 6 people, A, B, C, D, E, F be seated

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Manager
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In how many ways can 6 people, A, B, C, D, E, F be seated  [#permalink]

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07 Feb 2012, 21:34
1
8
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Difficulty:

75% (hard)

Question Stats:

51% (01:44) correct 49% (01:43) wrong based on 306 sessions

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In how many ways can 6 people, A, B, C, D, E, F be seated in a row such that C and D are not seated next to each other as well as A and B are not seated next to each other?

A 384
B 396
C 576
D 624
E 696

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Re: Permutations and Combinations  [#permalink]

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07 Feb 2012, 22:08
Smita04 wrote:
In how many ways can 6 people, A, B, C, D, E, F be seated in a row such that C and D are not seated next to each other as well as A and B are not seated next to each other?

A 384
B 396
C 576
D 624
E 696

This is very similar to a question I discussed in a post a week back. Let me give you the link of the post which explains the concept in detail. Perhaps you can check it out and see if you can figure out the answer of this one on your own! I have discussed two methods in it.

http://www.veritasprep.com/blog/2012/01 ... e-couples/
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Joined: 02 Sep 2009
Posts: 55277
Re: Permutations and Combinations  [#permalink]

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08 Feb 2012, 03:44
2
6
Smita04 wrote:
In how many ways can 6 people, A, B, C, D, E, F be seated in a row such that C and D are not seated next to each other as well as A and B are not seated next to each other?

A 384
B 396
C 576
D 624
E 696

Total # of arrangements of 6 people in a row is 6!;

Now, let's find # of arrangements where C and D as well as A and B ARE seated next to each other. Consider them as a single unit: {AB}, {CD}, E, F. Now, these 4 units can be arranged in 4!, but we should multiply this number by 2*2 as A and B, as well as C and D within their unit can be arranged in two ways: {AB} or {BA}, and {CD} or {DC}. Hence total for this case is 4!*2*2;

So, {# of arrangements}={total}-{restriction}=6!-4!*2*2=624.

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Re: In how many ways can 6 people, A, B, C, D, E, F be seated  [#permalink]

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08 Feb 2012, 05:18
Number of total arrangements = 6!
Restriction 1= AB & CD not next to each other --> let say AB and CD are considered as one unit, respectively
Restriction 2= AB is not the same as BA + CD is not the same as DC --> the number will increase by 2*2

Total number of arrangements - Number out of restrictions = Result

6! - (4!*2*2) = 720 - (24*2*2) = 624

Director
Joined: 23 Jan 2013
Posts: 549
Schools: Cambridge'16
Re: In how many ways can 6 people, A, B, C, D, E, F be seated  [#permalink]

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19 Apr 2015, 03:57
6 people in the row is 6!=720

glue method is best , so consider A&B and C&D sit together

now we have 4!=24, but 2 people in one pair and 2 people in another pair can sit in 4 positions, so 24*4=96

desirable condition is 720-96=624

D
Manager
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Posts: 104
Schools: ISB '18
GMAT 1: 680 Q49 V33
Re: In how many ways can 6 people, A, B, C, D, E, F be seated  [#permalink]

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16 Oct 2015, 00:22
VeritasPrepKarishma wrote:
Smita04 wrote:
In how many ways can 6 people, A, B, C, D, E, F be seated in a row such that C and D are not seated next to each other as well as A and B are not seated next to each other?

A 384
B 396
C 576
D 624
E 696

This is very similar to a question I discussed in a post a week back. Let me give you the link of the post which explains the concept in detail. Perhaps you can check it out and see if you can figure out the answer of this one on your own! I have discussed two methods in it.

http://www.veritasprep.com/blog/2012/01 ... e-couples/

@Veritasprep

What is the difference between this question and the one mentioned in the link provided? They seem to be the same to me so why the difference in the approach and consequently, different answers?
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Posts: 274
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Concentration: Marketing, Finance
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GPA: 3
Re: In how many ways can 6 people, A, B, C, D, E, F be seated  [#permalink]

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15 Jun 2017, 02:30
Bunuel wrote:
Smita04 wrote:
In how many ways can 6 people, A, B, C, D, E, F be seated in a row such that C and D are not seated next to each other as well as A and B are not seated next to each other?

A 384
B 396
C 576
D 624
E 696

Total # of arrangements of 6 people in a row is 6!;

Now, let's find # of arrangements where C and D as well as A and B ARE seated next to each other. Consider them as a single unit: {AB}, {CD}, E, F. Now, these 4 units can be arranged in 4!, but we should multiply this number by 2*2 as A and B, as well as C and D within their unit can be arranged in two ways: {AB} or {BA}, and {CD} or {DC}. Hence total for this case is 4!*2*2;

So, {# of arrangements}={total}-{restriction}=6!-4!*2*2=624.

Hi Bunuel,
In my opinion, this question translates into "neither A and B, nor C and D should sit together". I think that the correct answer is not mentioned in the options.
There are four cases possible

1. AB sit together and CD sit together
2. AB sit together but CD don't sit together
3. CD sit together but AB don't sit together
4. neither AB nor CD sit together.

We have to subtract 1,2, and 3 from total possible cases.

I have found the answer to be 336 in this manner.
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Posts: 27
GMAT 1: 720 Q50 V38
Re: In how many ways can 6 people, A, B, C, D, E, F be seated  [#permalink]

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12 Jul 2017, 01:11
ShashankDave wrote:
Bunuel wrote:
Smita04 wrote:
In how many ways can 6 people, A, B, C, D, E, F be seated in a row such that C and D are not seated next to each other as well as A and B are not seated next to each other?

A 384
B 396
C 576
D 624
E 696

Total # of arrangements of 6 people in a row is 6!;

Now, let's find # of arrangements where C and D as well as A and B ARE seated next to each other. Consider them as a single unit: {AB}, {CD}, E, F. Now, these 4 units can be arranged in 4!, but we should multiply this number by 2*2 as A and B, as well as C and D within their unit can be arranged in two ways: {AB} or {BA}, and {CD} or {DC}. Hence total for this case is 4!*2*2;

So, {# of arrangements}={total}-{restriction}=6!-4!*2*2=624.

Hi Bunuel,
In my opinion, this question translates into "neither A and B, nor C and D should sit together". I think that the correct answer is not mentioned in the options.
There are four cases possible

1. AB sit together and CD sit together
2. AB sit together but CD don't sit together
3. CD sit together but AB don't sit together
4. neither AB nor CD sit together.

We have to subtract 1,2, and 3 from total possible cases.

I have found the answer to be 336 in this manner.

Agreed ! The answer choices are incorrect. It should be 336.
Math Expert
Joined: 02 Sep 2009
Posts: 55277
Re: In how many ways can 6 people, A, B, C, D, E, F be seated  [#permalink]

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12 Jul 2017, 01:18
ShashankDave wrote:
Bunuel wrote:
Smita04 wrote:
In how many ways can 6 people, A, B, C, D, E, F be seated in a row such that C and D are not seated next to each other as well as A and B are not seated next to each other?

A 384
B 396
C 576
D 624
E 696

Total # of arrangements of 6 people in a row is 6!;

Now, let's find # of arrangements where C and D as well as A and B ARE seated next to each other. Consider them as a single unit: {AB}, {CD}, E, F. Now, these 4 units can be arranged in 4!, but we should multiply this number by 2*2 as A and B, as well as C and D within their unit can be arranged in two ways: {AB} or {BA}, and {CD} or {DC}. Hence total for this case is 4!*2*2;

So, {# of arrangements}={total}-{restriction}=6!-4!*2*2=624.

Hi Bunuel,
In my opinion, this question translates into "neither A and B, nor C and D should sit together". I think that the correct answer is not mentioned in the options.
There are four cases possible

1. AB sit together and CD sit together
2. AB sit together but CD don't sit together
3. CD sit together but AB don't sit together
4. neither AB nor CD sit together.

We have to subtract 1,2, and 3 from total possible cases.

I have found the answer to be 336 in this manner.

The language might be misleading here but still I translated "as well as" as "and" not as "or".
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Joined: 22 Sep 2016
Posts: 169
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GMAT 1: 710 Q50 V35
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Re: In how many ways can 6 people, A, B, C, D, E, F be seated  [#permalink]

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18 Jul 2017, 06:53
Bunuel wrote:
Smita04 wrote:
In how many ways can 6 people, A, B, C, D, E, F be seated in a row such that C and D are not seated next to each other as well as A and B are not seated next to each other?

A 384
B 396
C 576
D 624
E 696

Total # of arrangements of 6 people in a row is 6!;

Now, let's find # of arrangements where C and D as well as A and B ARE seated next to each other. Consider them as a single unit: {AB}, {CD}, E, F. Now, these 4 units can be arranged in 4!, but we should multiply this number by 2*2 as A and B, as well as C and D within their unit can be arranged in two ways: {AB} or {BA}, and {CD} or {DC}. Hence total for this case is 4!*2*2;

So, {# of arrangements}={total}-{restriction}=6!-4!*2*2=624.

What if AB are together, but CD are separate. Similarly, if CD are together and AB are separate.
These two cases shouls aldo be taken into account.
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Re: In how many ways can 6 people, A, B, C, D, E, F be seated  [#permalink]

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18 Jul 2017, 06:57
Hi Bunuel,
In my opinion, this question translates into "neither A and B, nor C and D should sit together". I think that the correct answer is not mentioned in the options.
There are four cases possible

1. AB sit together and CD sit together
2. AB sit together but CD don't sit together
3. CD sit together but AB don't sit together
4. neither AB nor CD sit together.

We have to subtract 1,2, and 3 from total possible cases.

I have found the answer to be 336 in this manner.[/quote]

The language might be misleading here but still I translated "as well as" as "and" not as "or".[/quote]

That's not correct. I have worked on similar problems and I agree what ShashankDave has to say. I am 100% sure about it. I ave spoken to various maths experts, and they've told me the same thing!

--== Message from the GMAT Club Team ==--

THERE IS LIKELY A BETTER DISCUSSION OF THIS EXACT QUESTION.
This discussion does not meet community quality standards. It has been retired.

If you would like to discuss this question please re-post it in the respective forum. Thank you!

To review the GMAT Club's Forums Posting Guidelines, please follow these links: Quantitative | Verbal Please note - we may remove posts that do not follow our posting guidelines. Thank you.

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Re: In how many ways can 6 people, A, B, C, D, E, F be seated   [#permalink] 18 Jul 2017, 06:57
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