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.