HSC Maths Extension 1 binomial theorem and combinatorics: deep dive (2026 guide)

What this guide covers

The binomial theorem and the underlying combinatorics are bread-and-butter Extension 1 topics. They appear in Section II most years, often as a 4-mark coefficient question or a 3-mark identity proof.

This guide covers:

• The fundamental counting principles (multiplication, addition, complementary).
• Permutations and combinations with worked examples.
• The binomial theorem in full, including the general term.
• Standard exam techniques: coefficient of $x^k$, term independent of $x$, identity proofs.
• Sum identities like $\sum \binom{n}{k} = 2^n$.

Counting principles

The multiplication principle

If a procedure has steps $1, 2, \dots, k$ and step $i$ can be done in $n_i$ ways (independent of the previous steps), the total number of ways is $n_1 \cdot n_2 \cdots n_k$.

The addition principle

If a procedure can be completed in one of several disjoint ways, the total is the sum of the counts for each way.

Complementary counting

$(\text{at least 1}) = (\text{total}) - (\text{none}).$

For "at least" and "at most" problems, the complement is often easier.

Permutations

The number of ways to arrange $n$ distinct objects in a row is $n!$. To arrange $r$ out of $n$: ${}^n P_r = \frac{n!}{(n - r)!}$.

For objects with repeats (say $n_1$ of one kind, $n_2$ of another), the number of distinct arrangements is

Circular arrangements of $n$ distinct objects: $(n - 1)!$ (fix one to break rotational symmetry).

Combinations

The number of unordered selections of $r$ from $n$ distinct objects is

Key identities:

• Symmetry: $\binom{n}{r} = \binom{n}{n - r}$.
• Pascal's rule: $\binom{n}{r} = \binom{n - 1}{r - 1} + \binom{n - 1}{r}$.
• Sum: $\sum_{r = 0}^{n} \binom{n}{r} = 2^n$.

The binomial theorem

For any non-negative integer $n$,

The general term is

So $T_1 = a^n$ (when $k = 0$) and $T_{n + 1} = b^n$ (when $k = n$).

Pascal's triangle

The first few rows:

Each row $n$ gives the coefficients of $(a + b)^n$.

Standard exam techniques

Find the coefficient of $x^k$ in IMATH_35

Use the general term: $T_{k + 1} = \binom{n}{k} (c x)^{n - k} d^k$. Wait, this gives $x^{n - k}$, not $x^k$. Adjust: for the power of $x$ to be $j$ in $(c x + d)^n$, we need $n - k = j$, so $k = n - j$. Coefficient: $\binom{n}{n - j} c^{j} d^{n - j} = \binom{n}{j} c^{j} d^{n - j}$.

Concrete example: coefficient of $x^3$ in $(2 x + 5)^7$.

$T_{k + 1} = \binom{7}{k} (2 x)^{7 - k} (5)^k$. For $x^3$: $7 - k = 3$, $k = 4$. Coefficient: $\binom{7}{4} (2)^3 (5)^4 = 35 \cdot 8 \cdot 625 = 175 \, 000$.

Find the term independent of IMATH_52

The general term $T_{k + 1}$ has some power of $x$. Set the power to $0$ and solve for $k$.

Example: find the term independent of $x$ in $\left( x^2 + \frac{3}{x} \right)^6$.

$T_{k + 1} = \binom{6}{k} (x^2)^{6 - k} \left( \frac{3}{x} \right)^k = \binom{6}{k} 3^k x^{12 - 2 k - k} = \binom{6}{k} 3^k x^{12 - 3 k}$.

Independent: $12 - 3 k = 0$, so $k = 4$. $T_5 = \binom{6}{4} 3^4 = 15 \cdot 81 = 1215$.

Sum identities

Several sum identities follow from the binomial theorem by substitution.

Identity

IMATH_63 . (Set $a = b = 1$.)

Identity

IMATH_65 for $n \ge 1$. (Set $a = 1$, $b = -1$.)

Identity

IMATH_69 . (Differentiate $(1 + x)^n$ and set $x = 1$.)

These are standard HSC identity-proof items. A typical question:

"Prove that $\binom{n}{0} + \binom{n}{2} + \binom{n}{4} + \dots = \binom{n}{1} + \binom{n}{3} + \binom{n}{5} + \dots$."

Subtract $\sum_{k = 0}^{n} (-1)^k \binom{n}{k} = 0$ from $\sum_{k = 0}^{n} \binom{n}{k} = 2^n$ to get that the alternating sum is $0$, then the even-indexed sum equals the odd-indexed sum, each being $2^{n - 1}$.

Common exam traps

Trap: Off-by-one in IMATH_77

$T_1$ has $k = 0$ ($a^n$), and $T_{n + 1}$ has $k = n$ ($b^n$). Many students get the indexing wrong, costing marks.

Trap: Wrong power-balance equation

For $\left( a x^p + b x^q \right)^n$, the power of $x$ in $T_{k + 1}$ is $p(n - k) + q k$. Not $p k + q(n - k)$.

Trap: Forgetting coefficients

In $\left( 2 x + 3 \right)^n$, the general term includes powers of both $2$ and $3$. Easy to forget.

Trap: Combinations vs permutations

Read carefully whether order matters. "A group of $5$" is unordered (combination); "a sequence of $5$" or "a queue" is ordered (permutation).

Trap: Repeated elements in permutations

Forgetting to divide by $r_1! r_2! \cdots$ for repeats massively over-counts.

Practice patterns

Pattern A: Coefficient of a specific power

Find the coefficient of $x^5$ in $(2 + 3 x)^8$. Find the coefficient of $x^{10}$ in $\left( x^2 + \frac{1}{x} \right)^{12}$.

Pattern B: Term independent of IMATH_99

Find the term independent of $x$ in $\left( 2 x - \frac{1}{x^2} \right)^9$ or $\left( x + \frac{2}{x^3} \right)^{12}$.

Pattern C: Identity proof using IMATH_103

Show that $\sum_{k = 0}^{n} \binom{n}{k} 2^k = 3^n$. (Set $x = 2$ in $(1 + x)^n = \sum \binom{n}{k} x^k$.)

Pattern D: Counting with constraints

How many $4$-letter words from MATHEMATICS contain at least one vowel? Use complementary counting from no-vowels.

Pattern E: Combined permutation/combination

In how many ways can $5$ books be chosen from $10$ and arranged on a shelf? $\binom{10}{5} \cdot 5!$, equivalently ${}^{10} P_5$.

Exam strategy

Binomial theorem questions are typically 2-4 marks each. Allocate $4-6$ minutes per question. Steps:

1. Write the general term $T_{k + 1}$ for the given expression.
2. Identify what is being asked: coefficient, specific term, independent term.
3. Form the right equation (matching power of $x$, etc.) and solve for $k$.
4. Substitute back to compute the answer.
5. State your answer clearly.

For identity proofs, the standard moves are: set specific values into $(1 + x)^n$, add or subtract two such substitutions, or differentiate and substitute. The library of identities you need is small.

Linked dot points

For more detail, see the dot point pages at permutations, combinations, binomial theorem and Pascal's triangle, and pigeonhole principle.

For NESA past papers and marking guidelines, refer to educationstandards.nsw.edu.au.