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How many combinations are possible (check image)?

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Jun 21, 2018


I got 960 but since that isn't one of the options, I think perhaps there are elements to the question that aren't in the image posted as the question.


Let's first look at the twins. We need them to sit together. There are 8 seats at the table, and so there are 8 places where the twins can be (seats 1, 2; 2, 3;...8, 1). In addition, we can have the brother on the right or the sister on the right, and so there are 2 ways they can sit in their seats. That's #8xx2=16# different ways for the twins to sit.

Now the uncle who can't sit next to the twins. So wherever the twins end up, that leaves 4 seats where the uncle can be. That means that there are #16xx4=64# different seating arrangements of the uncle and the twins.

Now we have the remaining 5 people to seat. There are 5 seats, and so we can seat them #5! = 120# ways.

This means that we have #64xx120 = 7680# ways to seat the people - but this assumes we are in a row and not in a circle.

Because we are at a round table, we don't have a "starting seat" or an "ending seat" - and so having the people arranged in seats 1 through 8 is the same as having the same arrangement in seats 2 through 1, and so on. And so we need to divide by the number of seats to rid ourselves of duplicates:


which doesn't match up with any of the choices listed. Perhaps there are elements to the question that weren't posted in the image?

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