2023 HSC Mathematics Extension 1
Question 10:
Combinatorics
A group with students and teachers is to be arranged in a circle.
In how many ways can this be done if no more than students can sit together?
How to recognise this question
Question type: combinatorics
- The command and notation point to combinatorics.
- The requested response is a multiple choice.
How to handle it: count a simple unrestricted set first, then add or subtract the arrangements that break the condition.
Watch out: Do not select a formula until its domain, interval, sign and units match the question.
Step-by-step answer
- With students and only teachers, the condition "no more than students together" forces every teacher-gap to be used and the student occupancies of the three gaps to be in some order.
- Arrange the distinct teachers in a circle: ways. This creates gaps.
- Choose which gap receives a single student: ways. Arrange the distinct students into the ordered gaps (sizes ): ways.
- Total: , since .
- Option A undercounts student permutations; C multiplies an extra beyond the circular teacher count already used; D over-factors pair internal orders separately from .
- Answer: B. (NESA multiple-choice key: B.)
Common mistake: choosing a familiar formula before checking that its conditions, signs, interval and units match this question.
The paper records 1 total mark for this group. Where the official marking guide provides useful partial-credit criteria, those criteria are stated in the steps above.
Try your hand at it again by answering these questions
Question 1
Three distinct people sit in a circle. The number of distinct circular arrangements is
Question 2
Three gaps must be filled with people. The number of ways to choose which gap gets person is
Question 3
If the distinct teachers were instead arranged in a line (not a circle), their arrangements would number