2020 HSC Mathematics Extension 1
Question 8:
Combinatorics
Out of contestants, six are to be selected for the final round of a competition. Four of those six will be placed 1st, 2nd, 3rd and 4th.
In how many ways can this process be carried out?
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
- First choose the six finalists unordered: .
- Then award ordered places 1st–4th among those six: .
- Product: .
- Option A is only the unordered selection; B overcounts ordered structure; D has the wrong factorial denominators.
- Answer: C.
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
From players, reach a final; then of those are ranked 1st–3rd. How many ways?
Question 2
If six of ten are selected and no further ranking occurs, the count is
Question 3
From contestants, are chosen for a final and all are then fully ranked. How many ways?