2024 HSC Mathematics Extension 1
Question 3:
Pigeonhole principle
Students from different schools come together to form a choir.
What is the minimum size of the choir to know that there must be at least students in the choir from one of the schools?
How to recognise this question
Question type: pigeonhole principle
- The command and notation point to pigeonhole principle.
- The requested response is a multiple choice.
How to handle it: identify the objects and boxes, then show why avoiding a repeated box is impossible.
Watch out: Do not select a formula until its domain, interval, sign and units match the question.
Step-by-step answer
- Pigeonhole principle (worst case): postpone the guarantee as long as possible by giving each school only students.
- Four schools students students, and still no school has .
- The next student — the th — forces some school to reach .
- In general the guarantee of at least in one of groups is .
- Here . Option A is the worst case without the guarantee; C and D overshoot.
- 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
What is the minimum number of students from schools guaranteeing at least from one school?
Question 2
Minimum size guaranteeing at least students from one of schools is
Question 3
To guarantee at least red marbles when drawing from a bag that only has red and blue, in the worst case you may first take red and all the blue. If there are only two colours and you want of one colour, the minimal guaranteeing monochromatic is