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# can anyone help?

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There are 8 people sitting at a circular table, including Sam, Rebecca, and Diane.  Find the number of ways of arranging the 8 people, so that Sam sits directly between Rebecca and Diane.

Jun 23, 2020

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First, think of Sam, Rebecca, and Diane as a "block" or one person. So there are now 6 people to consider (five other people and the "block"). There are $$\frac{6!}{6}$$ ways these 6 people can sit around a circular table. Six options for the first seat, five options for the second, etc. We divide by 6 because for one seating arrangement, we can rotate the table six times.

But we can't forget about the seating arrangments within the block. Rebecca can either be on the right side of Sam, and Diane is then on the left, or vice versa. So we have to multiply by two.

Solve for this to get our answer $$\frac{6!}{6}*2$$ = 240.

Jun 23, 2020