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AS Level · Pure Mathematics 2 WMA12

Binomial Expansion

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What You'll Learn

The Formula

How (a+b)^n expands into a sum using binomial coefficients and a pattern of powers.

Binomial Coefficients

What ^nCr means, how to calculate it, and why Pascal's triangle gives you the same answer.

Expansion Techniques

Expanding (a+bx)^n, finding specific terms, and handling negative values properly.

Real Applications

Finding coefficients of particular powers and solving for unknowns using expansion properties.

1. The Binomial Expansion Formula

What is Binomial Expansion?

A binomial expansion is a way to expand expressions like (a+b)ⁿ without multiplying out all the brackets. Instead of grinding through four multiplications to expand (3+2x)⁴, you use a formula that tells you exactly what each term will be.

The "binomial" part just means "two terms" — like a+b. The expansion part means breaking it down into a sum of simpler terms.

The Master Formula

Binomial Expansion Formula

(a + b)ⁿ = ⁿC₀ aⁿb⁰ + ⁿC₁ a^n-1b¹ + ⁿC₂ a^n-2b² + ... + ⁿC_n a⁰bⁿ

This looks intimidating, but there's a clear pattern once you spot it. Let's break it down:

The Pattern Inside

1. Binomial coefficients: Start at ⁿC₀, then ⁿC₁, ⁿC₂, etc. Each coefficient tells you what number multiplies that term.

2. Powers of a: Start at n, decrease by 1 each time: n, n–1, n–2, ..., 1, 0.

3. Powers of b: Start at 0, increase by 1 each time: 0, 1, 2, ..., n–1, n.

4. Total power in each term: Always adds up to n. For example, ⁿC₂ a^n-2b² has powers (n–2) + 2 = n. Always.

What is ⁿC_r (Binomial Coefficient)?

ⁿC_r is read as "n choose r". It's the number of ways to choose r items from n items. The formula is:

Binomial Coefficient Formula

ⁿC_r = n! / (r!(n–r)!)

where n! (n factorial) means n × (n–1) × (n–2) × ... × 2 × 1.

For example, 5! = 5 × 4 × 3 × 2 × 1 = 120.

Useful Shortcuts

ⁿC₀ = ⁿC_n = 1 — always. (You only choose either nothing or everything.)

ⁿC₁ = ⁿC_n-1 = n — the edges are always just n.

ⁿC_r = ⁿC_n-r — symmetry. This saves calculation.

Use your calculator: Most scientific calculators have a nCr button (sometimes labelled C or COMB). Press SHIFT + ÷ on many models.

Example: ⁵C₂

Calculate ⁵C₂

Using the formula:

⁵C₂ = 5! / (2! × 3!) = (5 × 4 × 3 × 2 × 1) / ((2 × 1) × (3 × 2 × 1))

Simplify:

= (5 × 4) / (2 × 1) = 20 / 2 = 10

Or use your calculator:

Type 5 nCr 2 → Answer: 10

✓ Try This

Calculate ⁶C₃ using the formula ⁿC_r = n! / (r!(n–r)!).

2. Pascal's Triangle — A Visual Alternative

What is Pascal's Triangle?

3. How to Expand Brackets Using Binomial Expansion

The Step-by-Step Process

one line per term, brackets everywhere, calculator for coefficients.

⁴

a b n

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Also in the full note

4. Applications: Finding Coefficients and Solving for Unknowns
What to Memorise
Concepts Checklist
Exam Tips & Common Mistakes
Pascal's Triangle vs. The Formula
Dealing with Negatives
Type 1: Finding the Coefficient of a Particular Term
Type 2: Finding an Unknown Parameter

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