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
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 an-2b2 has powers (n–2) + 2 = n. Always.
What is nCr (Binomial Coefficient)?
nCr is read as "n choose r". It's the number of ways to choose r items from n items. The formula is:
where n! (n factorial) means n × (n–1) × (n–2) × ... × 2 × 1.
For example, 5! = 5 × 4 × 3 × 2 × 1 = 120.
Useful Shortcuts
nC0 = nCn = 1 — always. (You only choose either nothing or everything.)
nC1 = nCn-1 = n — the edges are always just n.
nCr = nCn-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 nCr = 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.