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In how many different ways can 3 boys and 3 girls be seated in a row

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In how many different ways can 3 girls and 3 boys be seated at a recta  [#permalink]

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New post 30 Mar 2015, 11:35
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In how many different ways can 3 girls and 3 boys be seated at a rectangular table that has 3 chairs on one side and 3 stools on the other side, if two girls or two boys can never sit side by side?

A. 24
B. 36
C. 72
D. 84
E. 96
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Re: In how many different ways can 3 girls and 3 boys be seated at a recta  [#permalink]

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New post 30 Mar 2015, 12:59
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Hi Awli,

The prompt gives us the specific restriction that a boy can't sit next to another boy and a girl can't sit next to another girl. Since there are chairs on one side of the table and stools on the other side of the table, we have to account for 2 possible seating arrangements:

BGB
GBG

and

GBG
BGB

From here, we can use a simple permutation to get to the answer:

Moving from left-to-right....
For the first "spot", there are 3 different boys to choose from
For the second "spot", there are 3 different girls to choose from
For the third "spot", there then 2 different boys to choose from
For the fourth "spot", there are then 2 different girls to choose form
For the fifth and sixth "spots", we have the 1 boy and 1 girl that are left

(3)(3)(2)(2)(1)(1) = 36 possible seating arrangements for each of the two options.

(36)(2) = 72

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Re: In how many different ways can 3 girls and 3 boys be seated at a recta  [#permalink]

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New post 31 Mar 2015, 00:21
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Awli wrote:
In how many different ways can 3 girls and 3 boys be seated at a rectangular table that has 3 chairs on one side and 3 stools on the other side, if two girls or two boys can never sit side by side?

A. 24
B. 36
C. 72
D. 84
E. 96


Since 3 people will be on each side, and no two boys/two girls can sit together, on one side you will have Boy-Girl-Boy arrangement and on the other side you will have Girl-Boy-Girl arrangement.
From the 3 boys, choose 2 in 3C2 ways and a girl in 3C1 ways. Now you have split the boys and girls into BGB and GBG.
For BGB, pick a side (either chairs or stools) in 2 ways.
The girl takes the center seat and the 2 boys can be arranged around the girl in 2! ways.
On the other side, the boy sits in the center and the girls are arranged on his two sides in 2! ways.

Total arrangements = 3C2 * 3C1 * 2 * 2! * 2! = 72 ways.

Answer (C)
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Re: In how many different ways can 3 girls and 3 boys be seated at a recta  [#permalink]

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New post 07 May 2016, 08:51
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Awli wrote:
In how many different ways can 3 girls and 3 boys be seated at a rectangular table that has 3 chairs on one side and 3 stools on the other side, if two girls or two boys can never sit side by side?

A. 24
B. 36
C. 72
D. 84
E. 96


ans C
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Re: In how many different ways can 3 boys and 3 girls be seated in a row  [#permalink]

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New post 04 Jan 2017, 01:12
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3!*3!*2! = 6*6*2 = 72

Answer C
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Re: In how many different ways can 3 boys and 3 girls be seated in a row  [#permalink]

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New post 04 Jan 2017, 02:32
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Bunuel wrote:
In how many different ways can 3 boys and 3 girls be seated in a row of 6 chairs such that the girls are not separated, and the boys are not separated?

A. 24
B. 36
C. 72
D. 144
E. 288


Lets consider the group of all boys as one entity and group of girls as other entity

Now we have two entities to arrange which is possible in 2! ways

The group of boys within itself can be rearranged in 3! ways and
The group of Girls within itself can be rearranged in 3! ways

Total arrangements = 2!*3!*3! = 2*6*6 = 72

Answer: Option C
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Re: In how many different ways can 3 boys and 3 girls be seated in a row  [#permalink]

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New post 04 Jan 2017, 06:22
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Bunuel wrote:
In how many different ways can 3 boys and 3 girls be seated in a row of 6 chairs such that the girls are not separated, and the boys are not separated?

A. 24
B. 36
C. 72
D. 144
E. 288


Bunch up the boys together and the girls together. Now we have to arrange 2 groups which we can do in 2! ways.

The boys within themselves can be arranged in 3! ways and the girls within themselves can be arranged in 3! ways.

Total arrangements = 2 * 3! * 3! = 72

Answer (C)

For more on arrangements with constraints, check:
https://www.veritasprep.com/blog/2011/1 ... ts-part-i/
https://www.veritasprep.com/blog/2011/1 ... s-part-ii/
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Re: In how many different ways can 3 boys and 3 girls be seated in a row  [#permalink]

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New post 04 Jan 2017, 21:45
Condition: girls not separated..and boys are not separated

Count them as 1&1. Now we have 1boys n 1girl..

Arrangements =2!
And arrangements inside 3 girls & inside 3boys is=3! & 3!

Total=2!*3!*3! = 72 option C

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Re: In how many different ways can 3 boys and 3 girls be seated in a row  [#permalink]

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New post 06 Jan 2017, 10:57
Bunuel wrote:
In how many different ways can 3 boys and 3 girls be seated in a row of 6 chairs such that the girls are not separated, and the boys are not separated?

A. 24
B. 36
C. 72
D. 144
E. 288


We need to determine how many ways to arrange 3 boys and 3 girls in a row of six chairs when the boys must be seated together and the girls must be seated together. Since order matters, we have a permutation problem.

We can arrange the boys and girls as follows:

[B1)-B2)-B(3)] - [G(1)-G(2)-G(3)]

(Note: Since the boys must sit together and the girls must sit together, we have included the boys in one bracket and the girls in one bracket.)

Since we have 2 brackets, we can arrange those 2 brackets in 2! or 2 ways. However, we must account for the individual arrangements of the boys and the girls. The boys can be arranged in 3! or 6 ways and the girls can be arranged in 3! or 6 ways. Thus, the group can be arranged in 2 x 6 x 6 = 72 ways.

Answer: C
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Re: In how many different ways can 3 boys and 3 girls be seated in a row  [#permalink]

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New post 26 Mar 2018, 13:34
1
Hi Al,

There are a number of ways to do the "math in this question", but here's a way that you might find easy:

The fact that we have 6 chairs in a row means that we'll likely be doing a permutation. We're given the specific rule that the seating will have to be either GGGBBB or BBBGGG.

The first chair can have either a boy or a girl:

6 _ _ _ _ _

Whether it's a boy or a girl, the next two chairs must be the same gender:

6 2 1 _ _ _

Then the next 3 chairs must be the other gender:

6 2 1 3 2 1

Now multiply:

6x2x1x3x2x1 = 72 possibilities.

Final Answer:

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Re: 3 boys and 3 girls  [#permalink]

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