Couples

$3$ Scenarios

$(A)$ Each of the $5$ couples sit with partners facing each other. They can be seated in $5! = 120$ ways. All can exchange seats with partner so that's $120*2^5= 3840$ ways.

$(B)$ If one couple sits next to each other then there has to be another couple that sits across from them. The rest of the $3$ couples sit with partners facing opposite each other. The 2 couples that sit next to each other across the table are selected in $\binom{5}{2}=10$ ways. They can choose where to sit in $4$ ways. The couples can exchange places in $2$ ways. The other $3$ couples can choose their places in $3!= 6$ ways. Finally all of them can exchange places with partner in $2^5$ ways so that's $10*4*2*6*2^5= 15360$ ways.

$(C)$ If two couples sit next to each other side on one side of the table, two other couples must sit opposite them on the other side. The fifth couple then sits facing each other. The $4$ couples can be chosen in $\binom{5}{4}=5$ ways. They can choose their places in $3$ ways. They can sit on those seats in $4!= 24$ ways. All five couples can exchange seats with their partner in $2^5$ ways. So its $5*3*24*2^5=11520$ ways.

Total of $3840+15360+11520= 30720$ ways.

||||| |||-- ||--| |--|| --||| |---- --|-- ----|

These are the 8 ways the couples can be seated. | represents a couple sitting facing each other. -- represents a couple right next to each other. For each of the 8 combinations above, we can fit the couples in 5! ways. Each of the 5 couples can be seated in 2 ways in their slot. So the total is 8.5!.2^5 = 30720.