Ways of seating people around a circular table

**HelenaStage** · November 14th 2009, 06:33 AM

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?

**Soroban** · November 14th 2009, 07:47 AM

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

Duct-tape $A$ and $B$ together
. . Then we have 5 "people" to arrange: . $\boxed{AB}\;C\;D\;E\:F$
There are: . $4! = 24$ ways to seat them.

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

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

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

**Plato** · November 14th 2009, 11:21 AM

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.
$3(4!)=72$

**Soroban** · November 14th 2009, 11:28 AM

Hello, Plato!

Elegant solution! . . . I like it!

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