# Thread: Ways of seating people around a circular table

1. ## Ways 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?

2. 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.

3. Originally Posted by 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?
Here is a different and direct approach.
Seat A at the table. There are then three places to seat B.
The other four can be seated in 4! ways.
$\displaystyle 3(4!)=72$

4. Hello, Plato!

Elegant solution! . . . I like it!