# Thread: Arranging boys and girls in a line

1. ## Arranging boys and girls in a line

In how many ways 5 boys and 4 girls can be arranged such that exactly 2 girls (any) are together.

Here's my thinking;

5! * (C(4,2) * C(6,1) + C(2,1) * C(5,1) + C(4,1))

The boys can be arranged in any way. Group any 2 of the girls and they can choose from 6 different spots in line. Girl number 3 has 5 different spots to choose from. Then finally girl 4 has 4 spots to choose from.

Is this correct?

2. ## Re: Arranging boys and girls in a line

Originally Posted by Hanache
In how many ways 5 boys and 4 girls can be arranged such that exactly 2 girls (any) are together.
5! * (C(4,2) * C(6,1) + C(2,1) * C(5,1) + C(4,1))
The boys can be arranged in any way. Group any 2 of the girls and they can choose from 6 different spots in line. Girl number 3 has 5 different spots to choose from. Then finally girl 4 has 4 spots to choose from.
I am not sure why you are using + sighs?
The five boys create six positions to separate the girls.
There are $\dbinom{4}{2}$ ways to select two girls to stand together times 2 That is $12$.
There are $\dbinom{6}{1}$ ways for that pair to stand together.
Now there are $\dbinom{5}{2}$ times $2$ for the other two girls to stand separated .
The boys can stand $5!$ ways.

3. ## Re: Arranging boys and girls in a line

Originally Posted by Plato
I am not sure why you are using + sighs?
The five boys create six positions to separate the girls.
There are $\dbinom{4}{2}$ ways to select two girls to stand together times 2 That is $12$.
There are $\dbinom{6}{1}$ ways for that pair to stand together.
Now there are $\dbinom{5}{2}$ times $2$ for the other two girls to stand separated .
The boys can stand $5!$ ways.
wouldn't there be $\dbinom{8}{1}$ ways for the pair of girls to stand together? There are 9 total people. It looks like you treated it as 7.

4. ## Re: Arranging boys and girls in a line

Originally Posted by romsek
wouldn't there be $\dbinom{8}{1}$ ways for the pair of girls to stand together? There are 9 total people. It looks like you treated it as 7.
No this is a separation problem. The boys are the separators. Five separators create six positions to separate the girls.
But two girls are together and the other two are separated from each other as well as from the pair, that is a count of three.