Problem Solving

Hi,
This is a combinatorics problem that I'm looking for a similar approach to solving.(as I've done one problem below)
Three guys and three girls sit down in a row of six chairs. If no guy will sit next to another guy and no girl will sit next to another girl, in how many different ways can the guys and girls sit?

Another different problem asked me a similar problem where 6 people were sitting while two of them could not sit next to each other.
So I did 6! which gives me a combination of all possible ways.
Then I subtracted all the possible ways those two people could sit next to each other from 6!. This is done by the following method:
Assume that the two people Mr. J and Mr. M are one person. Then I have only 5 people sitting. So, that is 5! ways possible. The 5! has to be multiplied by 2 since JM and MJ both have to be accounted for.
So, the answer is 6!-(5!*2)

How do I do the guy, girl problem in this way? Please help out.
Thanks.
Villa

artivilla wrote:Hi,
This is a combinatorics problem that I'm looking for a similar approach to solving.(as I've done one problem below)
Three guys and three girls sit down in a row of six chairs. If no guy will sit next to another guy and no girl will sit next to another girl, in how many different ways can the guys and girls sit?

Another different problem asked me a similar problem where 6 people were sitting while two of them could not sit next to each other.
So I did 6! which gives me a combination of all possible ways.
Then I subtracted all the possible ways those two people could sit next to each other from 6!. This is done by the following method:
Assume that the two people Mr. J and Mr. M are one person. Then I have only 5 people sitting. So, that is 5! ways possible. The 5! has to be multiplied by 2 since JM and MJ both have to be accounted for.
So, the answer is 6!-(5!*2)

How do I do the guy, girl problem in this way? Please help out.
Thanks.
Villa

Best way to do permutations is to write down the arrangement.

BGBGBG - boys can be arranged in 3! ways
girls can be arranged in 3! ways

total 3! * 3! = 6*6 = 36

GBGBGB - girls can be arranged in 3! ways
boys can be arranged in 3! ways

total 3! * 3! = 6*6 = 36

36 + 36 = 72

Hope this helps.