Overview

The Binomial Theorem is one of the most important bridges between algebra and combinatorics.

It gives a formula for expanding expressions of the form

For small powers, we can expand directly:

and

The Binomial Theorem generalises this pattern to any nonnegative integer power $n$.

It explains:

• Why the coefficients in these expansions appear
• How to find a specific term without expanding everything
• How powers can be estimated or simplified
• How algebraic expansions connect to counting and probability

Motivation

Suppose we want to expand

Multiplying out eight copies by hand would be slow:

However, the Binomial Theorem lets us write the full expansion immediately.

It tells us that each term comes from choosing some copies to contribute $x$ and the remaining copies to contribute $2$.

For example, the term involving $x^5$ occurs when we choose $x$ from $5$ of the factors and $2$ from the other $3$ factors. The number of ways to choose which $5$ factors provide $x$ is

So the $x^5$ term is

This is the main idea of the theorem, expansion coefficients count choices.

Binomial Coefficients

Before stating the theorem, we recall the notation

This is read as $n$ choose $k$ and is defined by

It counts the number of ways to choose $k$ objects from a set of $n$ objects.

For example,

So there are $10$ ways to choose $2$ objects from $5$.

Useful special values include

and

Also,

This symmetry comes from the fact that choosing $k$ objects to include is the same as choosing $n-k$ objects to exclude.

The Binomial Theorem

For every nonnegative integer $n$,

Written out, this means

The general term is

Here, $k$ tells us how many times we choose $b$ from the $n$ factors.

Why the Formula Works

Consider

This means

with $n$ factors.

To form a term in the expansion, we choose either $a$ or $b$ from each factor.

Suppose we want a term containing exactly $k$ copies of $b$. Then we must choose $b$ from $k$ of the $n$ factors and $a$ from the remaining $n-k$ factors.

This produces

The number of ways to choose which $k$ factors supply $b$ is

Therefore, the coefficient of

is

This proves the Binomial Theorem.

Pascal's Triangle

The coefficients in binomial expansions form Pascal's Triangle:

The row beginning with $1,4,6,4,1$ gives the coefficients of

So

Each entry in Pascal's Triangle is the sum of the two entries above it. This reflects the identity

This identity is often useful in combinatorial arguments.

Expanding Powers

The most direct use of the Binomial Theorem is expanding powers.

Example 1

Expand $(x+3)^5$.

Using the Binomial Theorem,

The coefficients in row $5$ of Pascal's Triangle are

Therefore,

Example 2

Expand $(2x-y)^4$.

Here, take

and

Then

Now compute each term:

As such

Finding a Specific Term

Often we do not need the entire expansion. We only need one coefficient or one term.

The general term in

is

This is the term containing $b^k$.

Example 3

Find the coefficient of $x^7$ in $(x+2)^{10}$.

The general term is

We want the exponent of $x$ to be $7$, so

Thus

The coefficient is

Now

so the coefficient is

Example 4

Find the constant term in

$$\left(x+\frac{2}{x}\right)^6.$$

The general term is

This becomes

For the constant term, we need the exponent of $x$ to be $0$:

Thus

The constant term is

Alternating Signs

The Binomial Theorem also handles expressions of the form

Since

we get

So the signs alternate depending on the parity of $k$.

For example,

A common mistake is to get these signs wrong, especially in middle terms.

Part 2

Read the continuation in Part 2!