Hi Tony351,

Let's designate one of the men to be Fred and specify the seating relative to Fred's place at the table, numbering the seats counterclockwise as 1,2,3,4,5,6,7,8 with Fred in seat 1.

We can choose the woman opposite Fred in one of 4 ways. There are now 6 people left.

We can choose the person in seat 2 in one of 6 ways and the person opposite in one of 3 ways. There are now 4 people left.

We can choose the person in seat 3 in one of 4 ways and the person opposite in one of 2 ways. There are now 2 people left.

We can choose the person in seat 4 in one of 2 ways and the person opposite in only 1 way. We're done.

Putting it all together, the number of possible arrangements is

4 * (6 * 3) * (4 * 2) * (2 * 1) = 1152.