johnd32 wrote:
There are 2 ways that come to mind to solve this, but I'm getting different answers. I've been trying to figure this out for a while. Just can't see where I'm going wrong.
Option 1
If 3 of them are to not sit next to each other, those three can sit in the following set up. In the diagram below, Y denotes a seat filled with someone who does not want to sit next to another person, and _ denotes an empty seat which can be filled by those without seating restrictions.
Y _ Y _ Y _
Y _ _ Y _ Y
Y _ Y _ _ Y
_ Y _ Y _ Y
FOR EACH of the options above, the 3 of them can sit in 3! ways = 6 ways. In addition to that, FOR EACH of the options, the remaining empty seats can also be filled in 3! ways = 6 ways.
Hence, the ways the 3 of them won't sit next to each other is 4*6*6 = 144
Option 2
Calculate total number of seating arrangements - Ways 3 of them can sit together
Total number of seating arrangements = 6! = 720
For ways 3 of them can sit together, pretend the group is a group of 4 individuals instead of 6 (because 3 of them have to sit together anyway). If you had 4 individuals, you'd have 4! ways = 24 ways. However, one of those individuals is actually a group of 3 people, and the ways that group of 3 people can sit is 3! = 6. This 6 multiplied by the 24 calculated earlier gives you the number of ways 3 of them can sit together, which is 144.
Total seating arrangements - Ways of sitting together = 720 - 144 = 576
I'm getting different answers in Option 1 and Option 2, and I'm not sure if either of them is even correct. Any help would be appreciated. Thanks everyone!
The reason you are getting different answers for Option 1 and Option 2 is because each of those solutions are answering a different question. The question is not very clear on what is meant by "Rob, Ray, and Susan will not sit next to each other".
If this means no two of them can sit together (for instance, R-R-L-M-J-S is not acceptable), then the number of different arrangements is 144. Your first option is correctly answering this question.
If "Rob, Ray, and Susan will not sit next to each other" means that two of them can sit together as long as all three are not sitting together (i.e. R-R-L-M-J-S is acceptable), then the number of different arrangements is 576. Your second option is answering this question.
If you wanted to get the answer of 576 using the method of your first option, you would also have to take into account the arrangements such as YY_Y_ _, Y_YY_ _ etc.
If you wanted to get the answer of 144 using the method of your second option, you need to calculate the number of arrangements where at least two of the three people are sitting together. If we consider Rob and Ray sitting together, for instance, the number of arrangements is 2*5!. The same is true when Rob and Susan sit together, as well as when Ray and Susan sit together. When you add all these arrangements, you will have double counted the arrangements when all three of them sit together and we know the number of such arrangements is 3!*4!.
Thus, in 2*5! + 2*5! + 2*5! - 3!*4! = 6*5! - 6*4! = 6*4!(5 - 1) = 576 of the arrangements, at least two of the three people are sitting together and thus, in 720 - 576 = 144 of the arrangements, no two of those people are sitting together.