2022 HSC Mathematics Extension 1
Question 12(b):
Pigeonhole principle
A sports association manages 13 junior teams. It decides to check the age of all players. Any team that has more than 3 players above the age limit will be penalised.
A total of 41 players are found to be above the age limit.
Will any team be penalised? Justify your answer.
How to recognise this question
Question type: pigeonhole principle
- The command and notation point to pigeonhole principle.
- The requested response is a explanation.
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
- If no team were penalised, each of the 13 teams would have at most 3 over-age players, giving at most over-age players in total.
- But , so this is impossible.
- By the pigeonhole principle, at least one team has at least 4 over-age players and will be penalised.
- NESA: 2 marks correct; 1 mark for attempting the pigeonhole principle (or equivalent).
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
12 teams, more than 2 over-age players triggers a penalty, 25 over-age players total. Must a team be penalised?
Question 2
13 teams, penalty for more than 3, exactly 39 over-age players. Must a team be penalised?
Question 3
With teams and max allowed, how many over-age players guarantee a penalty?