Combinatorics (ME-A1)

4 dot points across 4 inquiry questions, in syllabus order. Each dot point has a focused answer, with exam-style questions and worked answers 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

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 read →

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

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 read →

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

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 read →

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

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 read →