2025 HSC Mathematics Extension 1
Question 13(b):
Combinatorics
Count circular seatings of eight people when two specified people may not sit together.
How to recognise this question
Question type: combinatorics
- The command and notation point to combinatorics.
- The requested response is a worked solution.
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
- Unrestricted circular arrangements: .
- If the two guests sit together, treat them as one block: circular arrangements, with internal orders.
- .
Common mistake: choosing a familiar formula before checking that its conditions, signs, interval and units match this question.
The paper records 2 total marks 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
Seven people sit around a round table. Two specified people must not sit together. How many arrangements?
Question 2
Choose a committee of 4 from 9 people if two specified people cannot both serve.
Question 3
Evaluate .