2024 HSC Mathematics Extension 1
Question 9:
Combinatorics
A bag contains metal coins, , that are made from either silver or bronze.
There are silver coins in the bag and the rest are bronze.
Two coins are to be drawn at random from the bag, with the first coin drawn not being replaced before the second coin is drawn.
Which of the following expressions will give the probability that the two coins drawn are made of the same metal?
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
- Without replacement, ordered draws: and .
- Add them: , which is option A.
- Option D uses in the denominator — that is the with-replacement (independent) model.
- Option C has the correct combination numerators but the wrong denominator: unordered sample space size is , not .
- Option B is dimensionally and combinatorially incoherent for this probability.
- Answer: A. (NESA multiple-choice key: A.)
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
For the same bag, the same-metal probability as an unordered selection of coins is
Question 2
If the first coin is replaced, becomes
Question 3
Bag has coins with silver. Without replacement, equals