2022 HSC Mathematics Extension 1
Question 11(c):
Combinatorics
Find the coefficients of and in the expansion of .
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
- Binomial general term: .
- For : . Coefficient of is .
- For : . Coefficient of is .
- NESA: 2 marks correct; 1 mark for one correct coefficient (or equivalent merit).
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
Find the coefficients of and in .
Question 2
Coefficients of and in .
Question 3
Write the term in without full expansion.