a. The first part asks for how many ways they can be seated together in a row. Therefore we want the permutations of the set of 6 people, or 6 factorial,
6! = 6 [tex]*[/tex] 5
= 30 [tex]*[/tex] 4
= 360 [tex]*[/tex] 2 = 720 possible ways to order 6 people in a row
b. There are two cases to consider here. If the doctor were to sit in the left - most seat, or the right - most seat. In either case there would be 5 people remaining, and hence 5! possible ways to arrange themselves.
5! = 5 [tex]*[/tex] 4
= 20 [tex]*[/tex] 3
= 120 [tex]*[/tex] 1 = 120 possible ways to arrange themselves if the doctor were to sit in either the left - most or right - most seat.
In either case there are 120 ways, so 120 + 120 = Total of 240 arrangements among the 6 people if the doctor sits in the aisle seat ( leftmost or rightmost seat )
c. With each husband on the left, there are 3 people left, all women, that we have to consider here.
3! = 3 [tex]*[/tex] 2 6 ways to arrange 3 couples in a row, the husband always to the left
