Arne, Bobbette, Chuck, Deirdre, Ed, and Fran have reserved six seats in a row at the theater, starting at an aisle seat. In how many ways can they arrange themselves if the men and women are to alternate seats and a man must sit on the aisle? Arne, Chuck, and Ed are the men, and the others are women (Hint: First answer the following six questions in order.)
(a) How many choices are there for the person to occupy the first seat, next to the aisle? (It must be a man.)
(b) How many choices of people may occupy the second seat from the aisle? (It must be a woman.)
(c) Now, how many for the third seat? (one of the remaining men)
(d) Now, how many for the fourth seat? (a woman)
(e) Now, how many for the fifth seat? (a man)
(f) Now, how many for the sixth seat? (a woman)
Now multiply your six answers above.
Answer
Final Answer: The total number of ways they can arrange themselves is \(\boxed{36}\).
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 \(\boxed{36}\).
