A couple sitting together at a round table

Question

Seven people sit around a round table (rotations count as the same seating). Two of them are a couple who insist on sitting right next to each other. How many distinct seatings satisfy this?

Accepted Answer

Glue the couple into a single block, so the two of them travel together as one unit. That leaves 6 units to arrange around the circle: the block plus the other 5 people. Circular arrangements of 6 units give (6-1)! = 5! = 120. But within the block, the couple can sit in 2 internal orders (person A on the left or on the right), so multiply: 5! 	imes 2 = 120 	imes 2 = 240. Takeaway. To force two people together in a circle, treat them as one block: arrange (n-1) units as (n-2)!, then multiply by 2 for the block's internal order. Here 5! 	imes 2 = 240.

A couple sitting together at a round table

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