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?
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.