Students from 4 different schools come together to form a choir.
What is the minimum size of the choir to know that there must be at least 20 students in the choir from one of the schools?
Combinatorics, EXT1 A1 2022 HSC 12b
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. (2 marks)
Combinatorics, EXT1 A1 2021 HSC 10 MC
The members of a club voted for a new president. There were 15 candidates for the position of president and 3543 members voted. Each member voted for one candidate only.
One candidate received more votes than anyone else and so became the new president.
What is the smallest number of votes the new president could have received?
- 236
- 237
- 238
- 239
Combinatorics, EXT1 A1 2020 HSC 12c
To complete a course, a student must choose and pass exactly three topics.
There are eight topics from which to choose.
Last year 400 students completed the course.
Explain, using the pigeonhole principle, why at least eight students passed exactly the same three topics. (2 marks)
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Combinatorics, EXT1 A1 EQ-Bank 14
A delivery company has 1095 packages to deliver on a given day.
It has 17 delivery vans that will deliver all packages. If one van delivers more packages than all other vans, the company pays the driver a $100 bonus.
What is the minimum number of packages a van could deliver and still win the $100 bonus. (2 marks)
Combinatorics, EXT1 A1 EQ-Bank 13
A sock drawer contains blue, white and green socks.
If individual socks are randomly chosen from the drawer, what is the minimum number that must be selected to ensure there are at least three pairs? (2 marks)
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Combinatorics, EXT1 A1 EQ-Bank 12
Eleven numbers are randomly chosen from the set of integers,
Prove that the sum of two of the eleven numbers randomly selected must equal 21. (2 marks)
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Combinatorics, EXT1 A1 SM-Bank 11
A multiple choice quiz asks students 4 questions. Each question has three possible answers, a, b or c, and students must attempt each question.
How many students must do the quiz to ensure that at least two sets of answers are identical? (2 marks)
Combinatorics, EXT1 A1 2010 HSC 7c
- A box contains
identical red balls and identical blue balls. A selection of balls is made from the box, where . Explain why the number of possible colour combinations is
. (1 mark)
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- Another box contains
white balls labelled consecutively from to . A selection of balls is made from the box, where . Explain why the number of different selections is
. (1 mark)
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- The
red balls, the blue balls and the white labelled balls are all placed into one box, and a selection of balls is made. Using the identity,
or otherwise, show that the number of different selections is . (3 marks)
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