Seating couples around 2 tables Here's my question and possible answer.
How many possible ways can you arrange 8 married couples between 2 circular tables of 8 identical chairs each such that:
1) each couple must sit at the same table, and,
2) at each table, men and women must sit in adjacent chairs (NOTE: a couple can sit next to each other but doesn't have to).
My solution:
Number of ways 
= (number of ways to split 8 couples into 2 tables of 4 couples each) * (number of arrangements at each table)
$=\frac{8!}{4!4!}*$(4 men and 4 women sitting alternately in 2 ways)
$=\frac{8!}{4!4!}\!\cdot\! 4!\!\cdot\!4!\cdot\!2$
$=2 * 8!$
I feel that I'm wrong about this. 
Can someone verify this solution or provide the correct one?
 A: First, pick $4$ couples out of the $8$ couples to sit at one table: $8 \choose 4$
Note that this will fix the people at the other table as well. Now, if we differentiate between the two tables, then the number of ways to split the $16$ people between the two tables is ${8 \choose 4} $ (that is the number of ways to pick the people for table 1, fixing the rest for table 2). If you do not differentiate between the tables, then divide this by $2$.
Now let's arrange the people. We'll calculate the number of ways to seat the people around one table, and just multiply by that number again for the other table at the end.
Since it's a circular table with identical chairs, we'll 'anchor' the seats with $1$ of the women. Then, we can seat the other women in $3!$ ways relative to this woman, and the men in $4!$ ways.
Total: $${8 \choose 4} \cdot 3! \cdot 4! \cdot 3! \cdot 4!$$
And again, if you do not differentiate between the tables, then divide this by $2$
A: How I would do it:
How many ways can you split the 8 couples into two groups of 4? That's ${8\choose4}=70$.
Then once split, you have 4 men, 4 women who need to be seated at one of the tables, in alternating order. The first person seated has 8 choices, then the other people of the same gender have 3,2,1 choices afterwards. Meanwhile for the opposite gender, they have 4,3,2,1 choices for their picks. So in total, $8\times4\times3\times3\times2\times2=1152$. Then to account for the rotational symmetry, we must divide by $8$ to give $144$ ways to arrange these couples on their table.
Then it is the same for the other table, so another $144$ ways to arrange them, hence $144^2$ ways to arrange all the people once tables have been chosen for each couple.
So the overall answer is $$70\times144^2=1,451,520$$
