800_or_bust wrote:Six people - A, B, C, D, E, and F - are being seated at a rectangular table with six seats. All of the seats are placed on the longest length of the table - three on one side, three on the other. No seats are placed on the short length of the rectangular table. If seats are selected at random, what is the probability that D is seated directly adjacent to F?
Here's how I thought about it:
The total number of possible arrangements is straightforward enough. 6 people and 6 seats, so 6!.
The number of desired outcomes is a little trickier. Imagine that D and F and sitting next to each other in the center and left seats on one side of the table. There'd be 4 more people to sit, and so
4! ways to place them. But D and F could also be in the center and right seats on that side of the table. So D and F can either sit center-left or center-right and there are two sides of the table, so there are
4 different places where D and F can sit. Last, we don't know if they're sitting D-F or F-D, so there are another
2! ways to sit. Desired: 4!*4*2!
Desired/Tota = (4!*4*2!)/6! = 4*2!/6*5 = [spoiler]4/15[/spoiler]
