Worked solution
Separate labelled circular seats from fixed-room colour assignments
(a) Five couples are to be seated at a round table with numbered seats. Find the number of ways such that each wife must sit next to her husband. marks
(b) Mr Lin wants to repaint his four bedrooms. Each room can only be painted with one colour. From six different colours, find the number of ways the four rooms can be painted if
(i) there is no restriction on the colours used, mark
(ii) there is no repetition of colours used, mark
(iii) he chooses exactly two colours. marks
[2011/MJC/II/7]
CJC · Review problems · Question 12
Answer
(a) 7680; (b)(i) 1296; (ii) 360; (iii) 210.
(a) Couple blocks with numbered seats
Arrange five couples circularly in ways and give each its internal spouse orders. Assign a reference person to one of 10 numbered seats. Thus .
(b)(i) Four independent room colours
Each of four distinct rooms has six choices, giving .
(b)(ii) Method 1: select, then assign colours
Choose four of the six colours and assign them to the four rooms: .
(b)(ii) Method 2: multiply the remaining choices
The four rooms have 6,5,4,3 choices in sequence, so . The source’s factor annotations refer to “doors” although the question is about bedrooms; the distinct objects being coloured remain the four rooms.
(b)(iii) Use both selected colours
Choose two colours in ways. Either one colour is used in three rooms and the other in one, giving assignments, or each is used in two rooms, giving . Thus . Selecting two colours is not sufficient unless both are actually used.
Common mistakes
Do not count a monochrome assignment after choosing two available colours.
Solution notes
All asks, both source no-repetition methods, complete two-colour cases and marks retained. Source “door” annotation reconciled with the bedroom prompt.