# Twelve guests are to be seated for dinner, six of them have already chosen

#### randy.lahey

##### New member
[FONT=&quot]Twelve guests are to be seated for dinner, six of them have already chosen to sit in a row at a bar table, and the other six will sit in a circle at a round table. What is the total number of different seating arrangements for these guests?[/FONT]

#### pka

##### Elite Member
WHY mention the guests sitting at the bar?
Because sitting at a bar is a linear arrangement. Whereas, sitting around a table is circular.

Twelve guests are to be seated for dinner, six of them have already chosen to sit in a row at a bar table, and the other six will sit in a circle at a round table. What is the total number of different seating arrangements for these guests?
$$\displaystyle _{12}\mathcal{P}_6~\cdot~5!$$ where $$\displaystyle _{12}\mathcal{P}_6$$ is the notation for a permutation of twelve things arranged six at a time.
@randy.lahey you need to reply to this with an explanation of the answer.

#### mmm4444bot

##### Super Moderator
Staff member
Doesn't that clearly mean that only 6 guests are involved?

The other 6 are staying at the bar...
There's another interpretation.

Six sit at the bar and six sit at the table; that's one arrangement for these 12 guests.

A second arrangement would occur, if the six who chose to sit at the bar were not the same six people in the first arrangement.

A third arrangement would occur, if the six who chose to sit at the bar were not the same six people in either the first or second arrangement.

And so on ...

For each of the arrangements above, the six people at the bar could sit down in a different order.

And then for each of all possible arrangements involving the six people at the bar, the remaining six could be seated in different orders around the circle.

The question is not clear because it references "these guests". Does "these" mean all 12 guests or something else? :cool: