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?
Hello, HelenaStage!
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 $\displaystyle B$ are adjacent.
Therefore, there are: .$\displaystyle 120 - 48 \:=\:72 $ ways in which $\displaystyle A$ and $\displaystyle B$ are not adjacent.