Lesson 20.2

Binomial Theorem & Pascal's Triangle

Expanding (a+b)ⁿ efficiently using combinatorics.

Introduction

Expanding by hand is tedious. The Binomial Theorem gives us a formula to find any term directly using binomial coefficients, which appear in the beautiful pattern known as Pascal's Triangle.

Prerequisite Connection

You understand factorials and basic exponent rules.

Today's Increment

We use to expand binomials and find specific terms.

Why This Matters

Binomial coefficients appear in probability, statistics, and polynomial algebra throughout mathematics.

Key Concepts

Binomial Theorem

Binomial Coefficient

Read as "n choose k" — the number of ways to choose k items from n.

Pascal's Triangle

1 1

1 2 1

1 3 3 1

1 4 6 4 1

Each entry = sum of two entries above it.

Worked Examples

Example 1: Full Expansion (Basic)

Expand

Row 4 of Pascal's Triangle: 1, 4, 6, 4, 1

Example 2: Finding a Specific Term (Intermediate)

Find the coefficient of in

Term with :

Coefficient = 720

Example 3: Negative Terms (Advanced)

Expand

Treat as

Common Pitfalls

Forgetting the coefficient

In , the 2 is also cubed: , not .

Sign errors with negatives

but . Track signs carefully!

Wrong term index

The (k+1)th term uses , not .

Real-World Application

Probability Distributions

The binomial distribution uses these coefficients! The probability of exactly k successes in n trials involves .