Sec 3 A-Math Binomial Expansions

Overview

What you’ll learn

Evaluate factorials and binomial coefficients, including the boundary cases (n 0) = (n n) = 1 and 0! = 1.
State the number of terms in the expansion of (a + b)ⁿ and connect the coefficients to Pascal’s triangle.
Expand (a + b)ⁿ for small positive integers n using the Binomial Theorem.
Write down the general term and use it to find a specific term or coefficient, such as the term independent of x.
Write the first few terms of an expansion in ascending (or descending) powers of x, simplifying coefficients fully.

In 30 seconds

1n! = n × (n − 1) × … × 2 × 1, and 0! = 1 by convention.
2Binomial coefficient: (n r) = n!/(r!(n − r)!); they are the rows of Pascal’s triangle: 1 4 6 4 1 for n = 4.
3(a + b)ⁿ has n + 1 terms; its expansion is Σ (n r) a^(n−r) b^r.
4General term: the term containing b^r is (n r) a^(n−r) b^r — the only formula you need to fish out one coefficient.
5In (a + bx)ⁿ the powers of b accumulate: the x² term carries b² — never forget to raise the b.
6A term “independent of x” has exponent 0: set the power of x to zero and solve for r.
7Signs alternate in (a − b)ⁿ: term r carries (−1)^r, so only even r can survive inside a positive coefficient question.

Notes · 1 of 5

Factorials

n! multiplies down to 1 — and the 0! = 1 convention oils the whole chapter.

For a positive integer n, the factorial n! multiplies every whole number from n down to 1: 4! = 4 × 3 × 2 × 1 = 24. The boundary convention 0! = 1 matters because it makes the coefficient formulas work at the ends of every expansion.

Factorial ratios simplify by cancellation rather than by computing both factorials: 7!/5! = (7 × 6 × 5!)/5! = 42. Questions rarely need a factorial larger than 10! written out.

Notes · 2 of 5

Binomial coefficients and Pascal’s triangle

(n r) chooses r of n things; the values line up in Pascal’s triangle.

The binomial coefficient (n r) = n!/(r!(n − r)!) counts how many ways r items can be chosen from n. The first few values are exactly the rows of Pascal’s triangle:

Pascal’s triangle
n	Row
0	1
1	1 1
2	1 2 1
3	1 3 3 1
4	1 4 6 4 1
5	1 5 10 10 5 1
6	1 6 15 20 15 6 1

Two boundary values come free: (n 0) = 1 and (n n) = 1 — there is exactly one way to choose nothing, or everything. Also note (n 1) = n and the symmetry (n r) = (n n − r), so (8 6) = (8 2) = 28.

Notes · 3 of 5

The Binomial Theorem

One formula expands (a + b)ⁿ for every positive integer n — no bracket-by-bracket grinding.

The Binomial Theorem says that for a positive integer n:

(a + b)ⁿ = (n 0) aⁿ + (n 1) aⁿ⁻¹b + (n 2) aⁿ⁻²b² + … + (n n) bⁿ.

1Count the terms
There are n + 1 terms: (a + b)⁷ has 8 terms.
2Walk the powers
The power of a steps down from n to 0 while the power of b steps up from 0 to n.
3Lay the coefficients
Take the coefficients from Pascal’s triangle row n, or compute each (n r).
4Simplify fully
Work out numeric factors — examiners want 40x³, not 10 × 4x³.

Notes · 4 of 5

The general term

The (r + 1)-th term formula finds any single term without expanding everything.

The term that contains b^r in the expansion of (a + b)ⁿ is called the general term:

term containing b^r = (n r) a^(n−r) b^r, for 0 ≤ r ≤ n.

1Identify a, b and n
In (2 + x)⁶: a = 2, b = x, n = 6.
2Match the requested power
A term in x⁴ needs r = 4.
3Substitute and simplify
(6 4) 2² x⁴ = 15 × 4 x⁴ = 60x⁴.

Notes · 5 of 5

Terms independent of x and first-few-terms questions

Set the net power of x to zero for the constant term; walk r = 0, 1, 2 for the opening terms.

A term independent of x has x-power 0. When both parts of the bracket contain x, write the general term’s total power of x and set it to zero — the resulting equation in r usually has one integer answer.

“First three terms in ascending powers of x” means r = 0, 1, 2 in order. For (1 + 3x)⁵: r = 0 gives 1; r = 1 gives (5 1)(3x) = 15x; r = 2 gives (5 2)(3x)² = 10 × 9x² = 90x². So the opening is 1 + 15x + 90x². Ascending questions usually stop at three or four terms — state “…” to show the expansion continues.

If the exponent equation for r has no integer solution — for example 12 − 3r = 5 — the expansion simply has no such term, and the coefficient is 0. Saying why earns the mark.

Revise

Key terms

The words examiners expect you to use precisely.

Factorial: The product n × (n − 1) × … × 2 × 1; with the convention 0! = 1.
Binomial coefficient: The value (n r) = n!/(r!(n − r)!), the r-th entry of Pascal’s triangle row n.
Binomial Theorem: The formula (a + b)ⁿ = Σ (n r) a^(n−r) b^r for positive integers n.
Expansion: The multiplied-out form of a bracket power, written as a sum of terms.
General term: The formula (n r) a^(n−r) b^r giving the term that contains b^r.
Coefficient: The numerical factor multiplying a power of x in an expansion.

Show 4 more key terms

Term independent of x: The term whose net power of x is 0 — a constant.
Ascending powers: Ordering terms from the lowest power of x to the highest.
Pascal’s triangle: The triangle of binomial coefficients in which each entry is the sum of the two above.
Exponent equation: The equation setting the net power of x to a requested value, solved for r.

Revise

Common mistakes

Where students most often lose marks in this chapter — and what to do instead.

Mistake: Treating 0! as 0.
The convention is 0! = 1, which is what makes (n 0) = 1 work in the formula.
Instead: 0! = 1; also (n 0) = (n n) = 1.
Mistake: Reading (n r) as n ÷ r or n^r.
The bracket notation is a binomial coefficient, not a power or a division.
Instead: (n r) = n!/(r!(n − r)!), e.g. (5 2) = 10.

Show 8 more mistakes

Mistake: Claiming (a + b)⁷ has 7 terms.
Each term corresponds to a power of b from 0 to 7 — that is 8 terms.
Instead: An expansion of (a + b)ⁿ has n + 1 terms.
Mistake: Forgetting to raise the second bracket: the x² term of (1 + 3x)⁵ written as 10 × 3x².
The b term appears to the power r, so (3x)² = 9x², and the coefficient is 90.
Instead: Write b^r fully before multiplying numbers.
Mistake: Using the general term (n r) a^r b^(n−r) with the powers swapped.
The power of a is n − r and the power of b is r; swapping answers questions about the wrong term.
Instead: Decide which bracket part is “b” (the one whose power you match), then use (n r) a^(n−r) b^r.
Mistake: Dropping the alternating signs in (a − b)ⁿ.
The minus sign belongs to b, so term r carries (−1)^r.
Instead: (3 − x)⁴ = 81 − 108x + 54x² − 12x³ + x⁴: signs alternate.
Mistake: Matching the wrong r, e.g. taking r = 4 for “the 4th term”.
The general term gives the term containing b^r — which is the (r + 1)-th term.
Instead: For “the k-th term”, use r = k − 1; for “the term in x^k”, match the power to k.
Mistake: Leaving coefficients unsimplified, e.g. answering 10 × 4x³ = “40x³” as 10(2²)x³.
Examiners award the answer mark for the simplified coefficient.
Instead: Multiply out every numeric factor before stating the term.
Mistake: Assuming every requested power exists, e.g. hunting x⁵ in (x² − 3/x)⁶ without checking r.
The exponent equation 12 − 3r = 5 has no integer solution, so there is no such term.
Instead: Solve the exponent equation for r first; if r is not an integer in range, the coefficient is 0.
Mistake: Adding factorial tails instead of cancelling: computing 9!/7! as a giant product.
It is slow and error-prone; the shared tail cancels.
Instead: 9!/7! = 9 × 8 = 72.

Check yourself

Quick quiz

18 questions with instant explanations. About 5 minutes.

Question 1 of 3

Question 1: current question
Question 2: not answered
Question 3: not answered

Evaluate factorials

What is the value of 4!?

Practise

Topical questions

Exam-style practice in three tiers, from recall to data questions.

5Foundation
5Standard
5Challenging

15 questions worth 60 marks, each with a model answer, mark scheme and examiner tips.

Practise topical questions

Exam practice

Practice paper

A 30-mark paper set out like a school paper, with a separate mark scheme. Give yourself 40 min.

Question paper

Chapter 4 Practice Paper

30 marks · 40 min · 5 pages

5 multiple-choice questions and structured questions with answer lines. Chapter 4 (A5 Binomial expansions).

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Mark scheme

PRO

Chapter 4 Practice Paper — mark scheme

Answers · Marking points · 3 pages

Model answers with a mark for each creditworthy point — for marking at home or in class.

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Where this chapter is tested

Each school sets its own scope — check your assessment outline.

Questions

Frequently asked questions

What does Sec 3 A-Math Chapter 4 cover?

It covers syllabus sub-topic A5 of O-Level 4049 / SEC G3 K341: factorials and binomial coefficients, the Binomial Theorem for positive integer n, expansions of (a + b)ⁿ, the general term formula, and finding specific coefficients including terms independent of x.

How do I find the term independent of x?

Write the general term and add up the total power of x across both bracket parts. Set that net power equal to zero, solve for r (it must be an integer between 0 and n), then substitute r back to get the constant term.

Why does my expansion of (1 + 3x)⁵ have the wrong x² coefficient?

Almost always because the 3 was not squared. The x² term is (5 2)(3x)² = 10 × 9x² = 90x² — the whole bracket 3x is raised to the power, not just the x.

How many terms should my expansion have?

The expansion of (a + b)ⁿ has exactly n + 1 terms. “First three terms” questions stop early on purpose; show the continuation with dots.

When is this chapter usually tested in school?

Binomial expansions typically appear in Weighted Assessment 2 (Term 2) and the end-of-year examination. Scope and weighting vary by school, so check your school’s assessment outline.

Sec 3 A-Math Binomial Expansions notes