Combinatorics (ME-A1)

4 dot points across 4 inquiry questions. Click any dot point for a focused answer with worked past exam questions where available.

How do we expand $(a + b)^n$ , and what does Pascal's triangle reveal about the coefficients?

State and use the binomial theorem, identify general and specific terms, and relate it to Pascal's triangle

A focused answer to the HSC Maths Extension 1 dot point on the binomial theorem. The expansion of $(a + b)^n$ using binomial coefficients, the general term $T_{k + 1}$ , applications to coefficient finding and approximation, and Pascal's triangle built row by row, with worked examples.

13 min answer →

How do we count the number of unordered selections (combinations) of objects from a larger set?

Use the combination formula

\binom{n}{r}

to count unordered selections, including with restrictions and complementary counting

A focused answer to the HSC Maths Extension 1 dot point on combinations. The combination formula, why you divide by $r!$ , key identities, applications including complementary counting, splitting into groups, and at-least/at-most counts, with a stepped diagram and worked examples.

13 min answer →

How do we count the number of ordered arrangements of a set of objects?

Use the multiplication principle and the permutation formula to count ordered arrangements, including restrictions and repeated elements

A focused answer to the HSC Maths Extension 1 dot point on permutations. The multiplication principle, the formula $\frac{n!}{(n - r)!}$ for arrangements of $r$ from $n$ , permutations of objects with repeats, circular permutations, and counting with restrictions, with stepped diagrams and worked examples.

13 min answer →

When can we guarantee that some box contains more than one object, and how do we apply this to counting and existence arguments?

State and apply the pigeonhole principle in counting and existence problems

A focused answer to the HSC Maths Extension 1 dot point on the pigeonhole principle. The basic and generalised forms, identifying boxes and pigeons in a problem, and using the principle to prove existence statements, with a stepped objects-into-boxes diagram and worked examples.

10 min answer →