Step 1 :First, we consider the first seat which must be occupied by a man. There are 3 men (Arne, Chuck, and Ed), so there are \(3\) choices for the first seat.
Step 2 :Next, we consider the second seat which must be occupied by a woman. There are 3 women (Bobbette, Deirdre, and Fran), so there are \(3\) choices for the second seat.
Step 3 :Then, we consider the third seat which must be occupied by one of the remaining men. There are 2 men left, so there are \(2\) choices for the third seat.
Step 4 :Next, we consider the fourth seat which must be occupied by one of the remaining women. There are 2 women left, so there are \(2\) choices for the fourth seat.
Step 5 :Then, we consider the fifth seat which must be occupied by the remaining man. There is 1 man left, so there is \(1\) choice for the fifth seat.
Step 6 :Finally, we consider the sixth seat which must be occupied by the remaining woman. There is 1 woman left, so there is \(1\) choice for the sixth seat.
Step 7 :To find the total number of arrangements, we multiply the number of choices for each seat. So, the total number of arrangements is \(3 \times 3 \times 2 \times 2 \times 1 \times 1 = 36\).
Step 8 :Final Answer: The total number of ways they can arrange themselves is \(\boxed{36}\).