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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
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.
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
_________________
Our greatest weakness lies in giving up. The most certain way to succeed is always to try just one more time.
I hated every minute of training, but I said, 'Don't quit. Suffer now and live the rest of your life as a champion.-Mohammad Ali
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....??
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.
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.
_________________
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...
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?
_________________
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...
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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73% (00:59) correct 
