Thread: Seating Around A Rectangular Table

Textbook: CRAWSHAW & CHAMBERS, "A Concise Course in A-level Statistics with worked examples" Third Edition.

I have a problem with the answers quoted in the book to two questions posed in Chapter 3. Exercise 3K, No 18, Pt(i) & Pt(ii). For those of you unfamiliar with the textbook, the problems concern seating around a rectangular table.

A rectangular table (two long & two short sides) has six seats arranged around it. One seat in the middle of each short side and two seats symmetrically arranged about the midpoint of each long side. The seats are identical. The questions are:

(i) how many different ways may six people (3 men, 3 women) be arranged around the table?

(ii) How many different ways are there to seat 3 men and 3 women alternately around the table.

The authors quote the answers as (i) 6! and (ii) 72

Are these answers not incorrect by a factor of two in each case because for each permutation the men/women may be moved three places clockwise or anti-clockwise and this will be the same arrangement?

The textboot is on its third edition so I'm wondering whether I've missed something that makes my analysis wrong.

Originally Posted by peterkelly01

A rectangular table (two long & two short sides) has six seats arranged around it. One seat in the middle of each short side and two seats symmetrically arranged about the midpoint of each long side. The seats are identical. The questions are:
(i) how many different ways may six people (3 men, 3 women) be arranged around the table?
(ii) How many different ways are there to seat 3 men and 3 women alternately around the table.
The authors quote the answers as (i) 6! and (ii) 72
Are these answers not incorrect by a factor of two in each case because for each permutation the men/women may be moved three places clockwise or anti-clockwise and this will be the same arrangement?

Those two answers are consistent with a linear understanding of this question. I would guess that authors would argue that because of the careful description of the table it is not circular.
I will agree with you that one could argue for a circular understanding of this.

If I were you, I would email the authors with this question.

Thanks for your comment. I would argue that circular or (as in this case) quasi-circular seating arrangements don't necessarily have to be associated with a circular table. For eaxmple, if the table were square and there was one seat at the mid-point of each side then this would (in my opinion, of course) have to be regarded as being a circular arrangement; provided, of course that the seated persons only have each other as points of reference. I like your idea re contacting the authors. The textbook is approximately 15 years old so hopefully they are still alive and well, and interested!!!

I did say that one could read the question as circular(quasi?).
But I suspect someone could argue that the table is located in a particular room. One end is near an outside window, another side is by a fireplace, ect. In other words, the table is oriented by the room and hence it is ordered. The authors could be thinking alone those lines. The given answers are correct for that reading.

That is why writing well posed questions is really difficult.

Yes, I agree. If the persons seated have points of reference beyond each other then the answers could be correct. I would term these as being "external" points of reference. When I see questions of this kind in testbooks, unless something contrary is stated, I assume that there are no such points of reference. It is akin to assuming the table to be located at the centre of a cubic space with the walls all painted the same colour, the table painted in one colour and the chairs absolutely identical. Perhaps I'm assuming too much!

I have seen questions concering seating around tables posed in a few school textbooks. The authors haven't defined what a circular seating arrangement actually is. They tend only to argue that there are (n-1)! ways of seating n persons around a circular table. Perhaps an author or would-be author will see this thread and address the problem in his/her next new book or book revision.