Worked solution
Use a manager as the circular reference
An office has 8 staff consisting of 5 employees,1 translator,1 secretary and 1 manager.
(i) The group stands in a line.
(a) Find the number of different possible arrangements. mark
(b) Find the number of different possible arrangements if the secretary and the manager do not stand together. marks
(ii) The group sits in a circle where the seats are indistinguishable.
(a) Find the number of different possible arrangements. mark
(b) Find the number of different possible arrangements if the secretary sits beside the manager, and the translator sits opposite the manager. marks
[2011/JJC/II/5]
CJC · Review problems · Question 9
Answer
(i)(a) 40320; (b) 30240. (ii)(a) 5040; (b) 240.
(i)(a) Arrange eight distinct people
.
(i)(b) Method 1: different gaps
Arrange the five employees and translator in ways. Their seven gaps include both ends. Choose two gaps for the manager/secretary and arrange that pair in ways: . The source calls these 6! arrangements “employees”; the sixth person is the translator.
(i)(b) Method 2: complement
If the manager/secretary form a block, there are arrangements. Subtract from all arrangements: .
(ii)(a) Identify rotations
Fix one person as reference. The unrestricted circular count is .
(ii)(b) Fix the manager and the opposite translator
There is one relative placement of the manager and translator, shown in the source as . The secretary has two neighbouring seats beside the manager (the source writes ). The five distinct employees fill the remaining seats in ways. Total . Reflection is not identified: the two sides of the manager give distinct arrangements.
Common mistakes
The five employees are distinct people, and the 6! gap construction includes the translator.
Solution notes
All four asks, both separation methods, source factor interpretations and printed marks retained. The source 6! “employees” label is explained to include the translator.