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M04-14

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M04-14  [#permalink]

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New post 16 Sep 2014, 00:22
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In how many ways 5 identical blue marbles and 6 identical green marbles can be arranged in a row, so that all the blue marbles are together?

A. 120
B. 30
C. 24
D. 11
E. 7

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New post 16 Sep 2014, 00:22
Official Solution:

In how many ways 5 identical blue marbles and 6 identical green marbles can be arranged in a row, so that all the blue marbles are together?

A. 120
B. 30
C. 24
D. 11
E. 7


There are 7 possibilities:

bbbbbgggggg

gbbbbbggggg

ggbbbbbgggg

gggbbbbbggg

ggggbbbbbgg

gggggbbbbbg

ggggggbbbbb

Formally, \(\frac{7!}{6!} = 7\).

Alternative explanation:

Think of all 5 blue marbles as a single unit. Together with 6 green marbles we'd have a total of 7 units. The difference between the arrangements is the position of the blue marbles (as a single unit). So the problem reduces to finding the number of unique patterns generated by changing the position of the blue marbles which can occupy 1 of 7 available positions. If the number of available unique positions is 7, then the number of unique patterns equals 7 as well.


Answer: E
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Re: M04-14  [#permalink]

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New post 12 Jun 2015, 11:20
Bunuel wrote:
Official Solution:

If a choir consists of 5 boys and 6 girls, in how many ways can the singers be arranged in a row, so that all the boys are together? Do not differentiate between arrangements that are obtained by swapping two boys or two girls.

A. 120
B. 30
C. 24
D. 11
E. 7


There are 7 possibilities:

bbbbbgggggg

gbbbbbggggg

ggbbbbbgggg

gggbbbbbggg

ggggbbbbbgg

gggggbbbbbg

ggggggbbbbb

Formally, \(\frac{7!}{6!} = 7\).

Alternative explanation:

Think of all 5 boys as a single unit. Together with 6 girls it makes a total of 7 units. The difference between the arrangements is the position of the boys (as a single unit). So the problem reduces to finding the number of unique patterns generated by changing the position of the boys who can occupy 1 of 7 available positions. If the number of available unique positions is 7, then the number of unique patterns equals 7 as well.


Answer: E


Hi Bunnel

What do we mean by the following statement-
Do not differentiate between arrangements that are obtained by swapping two boys or two girls.

Do we mean to say that we need to ignore the {5!} and {6!} ways in which the boys and girls can be arranged among themselves....??

This is what I understood.Please clarify.

Thanks
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Re: M04-14  [#permalink]

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New post 13 Jun 2015, 08:52
samichange wrote:
Bunuel wrote:
Official Solution:

If a choir consists of 5 boys and 6 girls, in how many ways can the singers be arranged in a row, so that all the boys are together? Do not differentiate between arrangements that are obtained by swapping two boys or two girls.

A. 120
B. 30
C. 24
D. 11
E. 7


There are 7 possibilities:

bbbbbgggggg

gbbbbbggggg

ggbbbbbgggg

gggbbbbbggg

ggggbbbbbgg

gggggbbbbbg

ggggggbbbbb

Formally, \(\frac{7!}{6!} = 7\).

Alternative explanation:

Think of all 5 boys as a single unit. Together with 6 girls it makes a total of 7 units. The difference between the arrangements is the position of the boys (as a single unit). So the problem reduces to finding the number of unique patterns generated by changing the position of the boys who can occupy 1 of 7 available positions. If the number of available unique positions is 7, then the number of unique patterns equals 7 as well.


Answer: E


Hi Bunnel

What do we mean by the following statement-
Do not differentiate between arrangements that are obtained by swapping two boys or two girls.

Do we mean to say that we need to ignore the {5!} and {6!} ways in which the boys and girls can be arranged among themselves....??

This is what I understood.Please clarify.

Thanks


Yes, you are correct.
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PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

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New post 14 Jul 2015, 13:34
Bunuel

I don't understand. I've seen any number of questions of this type. They are invariably permutations questions. If a bunch of people are put in a row then every arrangement, whether Paul is to the left of Michael or Michael is to the left of Paul, is different from every other arrangement.

In this case, the answer should be 5! x 7!
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New post 15 Jul 2015, 01:03
hessen923 wrote:
Bunuel

I don't understand. I've seen any number of questions of this type. They are invariably permutations questions. If a bunch of people are put in a row then every arrangement, whether Paul is to the left of Michael or Michael is to the left of Paul, is different from every other arrangement.

In this case, the answer should be 5! x 7!


Have you read this part of the question: Do not differentiate between arrangements that are obtained by swapping two boys or two girls.
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New post 15 Jul 2015, 09:57
I read that part but, to be honest, did not understand what it meant.
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New post 15 Jul 2015, 10:01
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New post 05 Aug 2015, 04:40
Bunuel wrote:
hessen923 wrote:
I read that part but, to be honest, did not understand what it meant.


It means that we are not interested in arrangements of girls and boys in their groups.



I think that should be re-worded in a clearer fashion. The fact that the problem mentioned two girls led me to look at permutation gbbbbbggggg as being able to rotate the first girl with any of the other remaining 5 girls. Apologies but the sentence " swapping two boys or two girls" does not make much sense to me...
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New post 20 Aug 2015, 08:43
CountClaud wrote:
Bunuel wrote:
hessen923 wrote:
I read that part but, to be honest, did not understand what it meant.


It means that we are not interested in arrangements of girls and boys in their groups.



I think that should be re-worded in a clearer fashion. The fact that the problem mentioned two girls led me to look at permutation gbbbbbggggg as being able to rotate the first girl with any of the other remaining 5 girls. Apologies but the sentence " swapping two boys or two girls" does not make much sense to me...


Edited the question. Is it OK now?
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M04-14  [#permalink]

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New post 30 Aug 2015, 08:46
Bunuel wrote:
CountClaud wrote:


I think that should be re-worded in a clearer fashion. The fact that the problem mentioned two girls led me to look at permutation gbbbbbggggg as being able to rotate the first girl with any of the other remaining 5 girls. Apologies but the sentence " swapping two boys or two girls" does not make much sense to me...


Edited the question. Is it OK now?


Much clearer. Thanks, Bunuel!
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New post 20 Jul 2016, 19:43
Should clarify to say "unique patterns." The problem attempts to do this by saying "identical" marbles. But this technically doesn't imply that marble 1,2,3,4,5,6,7 are interchangeable. I can move 2,1,3,4,5,6,7 and that seems like a different position to me, even if they are all the same size, weight, color etc. Unless, that's exactly what "identical" means on the GMAT, would be good to know if I am right or wrong. Logically, it doesn't seem like a strong enough implication per my example.
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New post 06 May 2017, 01:40
How is 7!/6! coming up??
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New post 15 Jun 2017, 05:39
Bunuel
If the question had been about unique Bs & Gs, then the answer should have been 7*5!*6!, correct?
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New post 15 Jun 2017, 06:38
kerin wrote:
Bunuel
If the question had been about unique Bs & Gs, then the answer should have been 7*5!*6!, correct?


No, in this case the answer would be 7!*5!.

Glue b's together, we'll get 7 units:
{bbbbb}{g}{g}{g}{g}{g}{g}

# of arrangements = 7!. 5 g's within their unit can be arranged in 5! ways. So, 7!*5!.
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Re: M04-14  [#permalink]

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New post 02 Oct 2018, 12:20
Question wording is wrong:

In how many ways 5 identical blue marbles and 6 identical green marbles can be arranged in a row, so that all the blue marbles are together?
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New post 03 Oct 2018, 00:19
"Do not differentiate between arrangements that are obtained by swapping two boys or two girls" is missing in the question.
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Re M04-14 &nbs [#permalink] 03 Oct 2018, 00:19
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