Worked solution
Seat ten people with two couples
A group of ten people consists of four single women, two single men and 2 couples. The ten people are seated randomly in a circle. Find the number of arrangements if (i) there is no restriction, (ii) the couples must seat together, (iii) the seats are of different colours.
CJC · Lecture · Self-Practice 6
(i) Fix one person as the circular reference
All ten people are distinct. With unlabelled circular seats, rotations represent the same arrangement, so .
(ii) Make one block for each couple
The two couple blocks and six single people give eight outer objects. Arrange these circularly in ways. Each couple has two internal orders independently, giving .
(iii) Distinguish all ten seat colours
Each seat now has its own identity. There are seat assignments, equivalently . This is a separate condition, so the couples need not remain together. All three printed answers are preserved; the derivations supplement the answer-only source.
Common mistakes
“The couples must seat together” means each pair sits together, not one block containing all four people. Each part imposes its own condition. In part (iii), seat colours distinguish rotations.