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If a choir consists of 5 boys and 6 girls, in how many ways can the [#permalink]

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10 Oct 2007, 23:36

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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.

Re: If a choir consists of 5 boys and 6 girls, in how many ways can the [#permalink]

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11 Oct 2007, 00:20

IrinaOK wrote:

The 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? (assume all members of the group are uniform and combinations within the group do not matter).

A 120 B 30 C 24 D 11 E 7

E. 7 Since members are uniform.

Let [B] be a cluster of boys such that all 5 are next to eachother.

_G_G_G_G_G_G_

We see that there is exactly 7 places for [B] to go (marked as an underscore [_])

Re: If a choir consists of 5 boys and 6 girls, in how many ways can the [#permalink]

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11 Oct 2007, 04:16

I have seen this question before, but it could be 7 true, but it could be higher if only the group of boys is to be counted as a group. If the girls can be rearranged then...it will be higher

Re: If a choir consists of 5 boys and 6 girls, in how many ways can the [#permalink]

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09 Jan 2008, 18:35

Got 7 as well. I thought about it like this:

_ _ _ _ _ _ _ _ _ _ _

There's 11 spots to fill, and the five boys have to be together. Start from the left and take up the first 5 spots: thats one way. Then, move one spot to the right, and take up the next 5, thats two ways. Keep going until you run out of space and youll see there are only 5 ways.

One thing to note is that the stem said that a different arrangements dont matter. If it did, we'd have many more ways, since in each 'way', the boys can be arranged in 5! ways.

Re: If a choir consists of 5 boys and 6 girls, in how many ways can the [#permalink]

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09 Jan 2008, 19:57

IrinaOK wrote:

The 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? (assume all members of the group are uniform and combinations within the group do not matter).

A 120 B 30 C 24 D 11 E 7

I couldnt figure out how to do this via combinatorics equations so I just wrote it down.

BBBBBGGGGGG. Had bout 30sec left and it dawned on me that all we have to realize is that we can arrange it

GBBBBBGGGGG, GGBBBBBGGGG etc... I count 7. U dont have to write all of these out just realize u can have 5 inbetween G's and 2 on the outside.

Re: If a choir consists of 5 boys and 6 girls, in how many ways can the [#permalink]

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28 Sep 2009, 11:28

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This post received KUDOS

The 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? (assume all members of the group are uniform and combinations within the group do not matter).

A 120 B 30 C 24 D 11 E 7

Soln. Since combinations within the group not matter, and since all boys go together. We can group all 5 boys into 1 group. The 6 girls and the 1 group of boys can be arranged in 7 ways.

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

M04-14

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.

Re: If a choir consists of 5 boys and 6 girls, in how many ways can the [#permalink]

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24 Sep 2016, 10:00

two boys (or two girls) are not different. see this question as the arrangement of 5 letters B and 6 letters G.

condition - letters B are together.

[BBBBB] G G G G G G --> so we have 7 elements (bracket as one element) here. arrangement for these 7 elements is 7! Now 6 out of these 7 elements are same i.e. G. in such case we divide by 6!

rule - in the arrangement of n elements if there are p,q,r .. are similar elements then arrangements are - n!/(p!*q!*r!...)

hence arrangement in above case is - 7!/6!

what about the elements in bracket? yes so arrangements for 5 B's are 5! BUT 5 elements are same so we divide by 5!. so essentially arrangements are = 5!/5!

Total Arrangements = (7!/6!) * (5!/5!) = 7 _________________

Re: If a choir consists of 5 boys and 6 girls, in how many ways can the [#permalink]

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26 Oct 2017, 01:06

Bunuel - I did figure out the question but i am still confused with the clause that has been mentioned in the stem :Do not differentiate between arrangements that are obtained by swapping two boys or two girls:

Assuming this was would not had been provided in the stem would the answer still be 7? or would it jump to 7*5! (5! being the arrangement for the boys internally).

Re: If a choir consists of 5 boys and 6 girls, in how many ways can the [#permalink]

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28 Oct 2017, 07:41

Hello everyone,

struggling a bit on figuring out the right approach to solve problems based on combinations.

My first approach here was to calculate the number of possible arrangements considering the "anagram grid" formula, i.e. 11! / (5! x 6!) - which is clearly wrong, but I can't really grasp why. What would I be calculating in this way?