Solution: This problem is a classical concept problem and can be solved using the permutation method. Suppose event The couple happens to be sitting together .

　　Method 1:　 people randomly sit around a round table. There are a total of methods. First consider the couple with the man on the left and the woman on the right sitting together: treat the two adjacent seats as a special seat and consider the idea of the bundling method. There are ways of arranging seats . In the same way, consider the man on the right and the woman on the left. Sitting method, so

.

　　Method 2　Only considers a couple. There are number of ways for a couple to sit randomly. Number the seats in row . The couple sits next to each other and on the right side of the man. There are ways to sit: the man sits in seat ; the woman sits in seat . Similarly, if the woman sits on the left side of the man, there are ways to sit. There are ways to sit in total, so 

.

　　Method 3　Assume that one person in the couple is sitting still and considering the other person (maybe it is a woman). This person sits randomly and there are ways to sit. If the couple is next to each other, she can only sit in the two positions to the left and right of the man, so

.