2023HSCShort answer

Q12 (d)

2 marks

Question

It is known that

\binom{n}{r}=\binom{n-1}{r-1}+\binom{n-1}{r}

for all integers such that $1\le r\le n-1$ . (Do NOT prove this.)

Find ONE possible set of values for $p$ and $q$ such that

\binom{2022}{80}+\binom{2022}{81}+\binom{2023}{1943}=\binom{p}{q}.

Try the question before revealing the answer.

Solution

Criteria	Marks
Provides correct solution	2
Uses given result to combine the first two terms on the left-hand side, or equivalent merit	1

Sample answer:

\begin{aligned} {}^{2022}C_{80}+{}^{2022}C_{81}+{}^{2023}C_{1943}&={}^{2023}C_{81}+{}^{2023}C_{1943}\\ &={}^{2023}C_{81}+{}^{2023}C_{80}\\ &={}^{2024}C_{81} \end{aligned}

So $p=2024$ and $q=81$ is a possible solution.

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Syllabus alignment

Combinatorics

Working with CombinatoricsApply permutations, combinations and counting principles to problems involving arrangements and selections.

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