# Combinatorics problems

silenzer

## Homework Statement

12 subjects (6 male, 6 female) are put on 3 lines, 4 in each. In how many ways can this be done, if one of the males and one of the females want to be on the same line?

None.

## The Attempt at a Solution

I thought it like this. I can pick the first 10 spots and leave the lovebirds behind. That can be done in 2 * (4C2) * 4! * 4! = 6912. 2*(4C2) because I'm picking spots for the two on the line where the lovebirds are missing, and they can be switched places so I multiply by 2. 4! * 4! is for the normal lines.

When I add the lovebirds to this, I get 6912 * 3 * 2 = 41472 because the unfinished line could be any of the 3, and multiply by 2 because, for each possibility in 4C2 above, I could have just put the couple there and switched them inwards.

Staff Emeritus
Homework Helper
If you didn't have the requirement that the lovebirds sit in the same line, there'd be 12!=479001600 ways to arrange the subjects, right? It seems that requiring two of them sit together shouldn't result in such a drastic reduction in possibilities.

## Homework Statement

12 subjects (6 male, 6 female) are put on 3 lines, 4 in each. In how many ways can this be done, if one of the males and one of the females want to be on the same line?

None.

## The Attempt at a Solution

I thought it like this. I can pick the first 10 spots and leave the lovebirds behind. That can be done in 2 * (4C2) * 4! * 4! = 6912. 2*(4C2) because I'm picking spots for the two on the line where the lovebirds are missing, and they can be switched places so I multiply by 2. 4! * 4! is for the normal lines.

When I add the lovebirds to this, I get 6912 * 3 * 2 = 41472 because the unfinished line could be any of the 3, and multiply by 2 because, for each possibility in 4C2 above, I could have just put the couple there and switched them inwards.
I don't understand the factor of 3. Any of the three what?

Don't you need to account for who sits in which rows still?

Jolb
Edit: vela has a better response than me and I can't seem to delete posts any more for some reason.

Staff Emeritus
Homework Helper
I got 130636800 ways. I might be off by a factor of 3, though.

It's okay to give out the answer in a problem like this. It's just not okay to show explicitly how you got the answer. It's the OP's job to figure out how to get the answer.

Jolb
Edit: I've gotten so confused with this problem that I want to edit this post even though vela has already responded. My fault.

Vela started his argument by saying there are 12! ways to arrange these people, but that does not try to eliminate double-counting configurations such as:
(A B C D) (E F G H) (I J K L) and (E F G H) (A B C D) (I J K L)
So this problem is ambiguous about whether or not those two cases are equivalent. So, OP, do you know whether the lines are "distinguishable" or not?

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Staff Emeritus
Homework Helper
I didn't consider those equivalent. Why do you assume they are?

Jolb
Well I guess there is ambiguity in the question. It doesn't say "Line A, Line B, Line C" so you could assume they are "indistinguishable" lines.

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Staff Emeritus