help!!please, (permutation)?

Question

there are 6 couples,(1) how many ways can the couples sit in the row.
(2)how many ways can the people sit alternately by gender?

Is it possible if (1)-> C(12,2) *C(10,2)*C(8,2) .....*C(2,2)
and (2) -> C(6,1)*C(6,1)*C(5,1)*C(5,1)*...*C(1,1)C(1,1)

Answer 1

(1) Assuming each couple sits together, then there are 3! ways that they can sit in the row, the number of ways to permute 3 objects.

(2) You can either have the men sit in positions 1,3,5 and the women in positions 2,4,6, or the other way around. If the men are in positions 1,3,5, then there are 3! = 6 ways to arrange them in those positions, and 6 ways to arrange the women as well, so 36 ways to arrange everyone, if the men are in the odd positions. Double that to account for when the men are in the even positions, and you get 72.

Answer 2

Rohit can't be right... note that his/her answer to (2) is greater than (1) which is impossible given (2) has more restrictions.

Here's the correct response. I'm assuming everybody has to sit right next to their spouse...

(1) First, let's count how many ways you can arrange within couples - regardless of how they are seated between themselves. Each couple can sit two ways: the man at the left of the woman (assuming hetero without loss of generality...) or the man at the right of the woman. This is independent of the other couples, so you can have 2^6 = 64 different combinations of within-couple arrangements.

For each of these 64 arrangements, there are 6! ways of seating the couples by assigning each couple to a pair of seats. So the answer is 64 x 6! = 46,080.

(2) Now, we have reduced the possible within-couple arrangements from 64 to just 2 (all men to the right of their woman, or all men to the left of their woman. So, 2 x 6! = 1,440.

Answer 3

1)If the couples sit together they can sit in 6! ways that is 120
2)If we fix the place for men (for ex:-1,3,5,7,9,11)then the ladies can be seated in the left out positions in i.e 2,4,6,8,10,12 in 6! ways but we can also keep the position of men fixed at 2,4,6,8,10,12 and that way the first will be a lady in the row,so that another 6!.so the answer is 2*6!.