Six people sit around a circular table. Person A and B must not sit next to each other...how many ways are there of seating them?

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- Nov 14th 2009, 06:33 AMHelenaStageWays of seating people around a circular table
Six people sit around a circular table. Person A and B must not sit next to each other...how many ways are there of seating them?

- Nov 14th 2009, 07:47 AMSoroban
Hello, HelenaStage!

Quote:

Six people sit around a circular table.

Person A and B must not sit next to each other.

How many ways are there of seating them?

With 6 people around a circular table,

. . there are: .$\displaystyle 5! \:=\:120$ ways to seat them.

Duct-tape $\displaystyle A$ and $\displaystyle B$ together

. . Then we have 5 "people" to arrange: .$\displaystyle \boxed{AB}\;C\;D\;E\:F$

There are: .$\displaystyle 4! = 24$ ways to seat them.

But the 5 "people" could be: .$\displaystyle \boxed{BA}\;C\;D\;E\;F$

Hence, there are: .$\displaystyle 2 \times 24 \:=\:48$ ways that $\displaystyle A$ and $\displaystyle B$adjacent.*are*

Therefore, there are: .$\displaystyle 120 - 48 \:=\:72 $ ways in which $\displaystyle A$ and $\displaystyle B$ areadjacent.*not*

- Nov 14th 2009, 11:21 AMPlato
- Nov 14th 2009, 11:28 AMSoroban
Hello, Plato!

Elegant solution! . . . I like it!