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How many combinations are possible (check image)?

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Jun 21, 2018

Answer:

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.

Explanation:

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:

#7680/8=960#

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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