---
title: "The Binomial Expansion"
description: "Expand (a + b)ⁿ with coefficients from Pascal's triangle or the nCr key, pick out a single term without writing the rest, and use the first few terms as an approximation when one term is small."
canonical: https://lightmysky.com/learn/mathematics/the-binomial-expansion-mt_z16HZkNZ8c
source: https://lightmysky.com/learn/mathematics/the-binomial-expansion-mt_z16HZkNZ8c.md
retrieved: 2026-09-12
---

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# The Binomial Expansion

Expand (a + b)ⁿ with coefficients from Pascal's triangle or the nCr key, pick out a single term without writing the rest, and use the first few terms as an approximation when one term is small.

Subject: Mathematics · Area: Algebra · Ages 16 to 17
Page: https://lightmysky.com/learn/mathematics/the-binomial-expansion-mt_z16HZkNZ8c

## Ready when they can

- Expand (1 + 2x)⁵ in full
- Find the coefficient of a stated power without expanding everything
- Approximate 1.02⁸ from the first three terms and comment on the error

## Lesson: Expanding brackets without the pain

The theorem expands a plus b to the n with no repeated multiplication. Powers of a fall from n to 0 while powers of b climb from 0 to n. Each term carries n choose k. A power of n always gives n plus 1 terms, which catches missing term slips.

**Example.** Row 5 of the triangle reads 1, 5, 10, 10, 5, 1. Pair each with the right power of 2x to expand 1 plus 2x to the 5. The full form is 1 plus 10x plus 40x squared plus 80x cubed plus 80x to the 4 plus 32x to the 5. Each entry is the sum of the two above it.

Single terms need no full expansion. The term with b to the k carries n choose k. For the x cubed term in 1 plus 2x to the 5, take 10 times 2 cubed. That gives a coefficient of 80. Track the powers of the second term with care.

**Tip.** First terms approximate when one part is small. For 1.02 to the 8, use 1 plus 8 times 0.02 plus 28 times 0.02 squared. That is 1.1712. Dropped terms are tiny but positive, so the estimate sits just below the true value.

**Recap.** Pair each coefficient with falling and rising powers, pluck single terms with n choose k, and approximate with the first terms.

## Practice

14 questions on this page, each with its working shown.

## Needs first

- [Counting Outcomes with the Product Rule](https://lightmysky.com/learn/mathematics/counting-outcomes-with-the-product-rule-mt_74gSbYIG8p)
- [Geometric Series and the Sum to Infinity](https://lightmysky.com/learn/mathematics/geometric-series-and-the-sum-to-infinity-mt_HHxFOv6vXS)

## Opens up

- [Binomial Coefficients and Combinatorial Identities](https://lightmysky.com/learn/mathematics/binomial-coefficients-and-combinatorial-identities-mt_lHXFvudTh6)
- [Taylor and Maclaurin Series](https://lightmysky.com/learn/mathematics/taylor-and-maclaurin-series-mt_SgI9Pn9RIO)
- [The Binomial Distribution](https://lightmysky.com/learn/mathematics/the-binomial-distribution-mt_X2kVGK67Md)