Binomial Theorem Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Binomial Theorem.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The binomial theorem gives the expansion of (a + b)^n as a sum of terms involving binomial coefficients: (a+b)^n = sum of C(n,k) * a^(n-k) * b^k. Each coefficient $\binom{n}{k}$ counts the number of ways to choose $k$ copies of $b$ from $n$ factors.

Each term of $(a+b)^n$ picks ' $a$ ' or ' $b$ ' from each factor. $\binom{n}{k}$ counts how many ways to pick $k$ $b$ 's.

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How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: The binomial theorem expands $(a+b)^n$ by counting, with $\binom{n}{k}$ , how many ways each term arises.

Common stuck point: The procedure for binomial theorem is the easy part; the trap is distributing the exponent as $(a+b)^n=a^n+b^n$ . Asking "Am I raising a two-term expression to a whole-number power and want its expansion or a single term?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I raising a two-term expression to a whole-number power and want its expansion or a single term?

Worked Examples

Example 1

medium

Expand

(x + 2)^3

using the Binomial Theorem.

Answer

x^3 + 6x^2 + 12x + 8

First step

Step 1: Apply

(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3

with

a = x

b = 2

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Example 2

hard

Find the coefficient of

x^3

in the expansion of

(2x + 3)^5

Example 3

medium

Find the coefficient of

x^4

(1+x)^7

Example 4

hard

Use binomial expansion to find the coefficient of

x^5

(1+x)^4(1+x)^6

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Expand

(a + b)^4

using Pascal's triangle.

Example 2

medium

What is

\binom{6}{2}

Example 3

easy

Compute the binomial coefficient

\binom{4}{2}

Example 4

easy

Expand

(a+b)^2

Example 5

easy

Expand

(a+b)^3

Example 6

easy

What is the coefficient of

a^2b

in the expansion of

(a+b)^3

Example 7

easy

Write the coefficients in the expansion of

(a+b)^4

Example 8

easy

Compute

\binom{5}{2}

Example 9

easy

How many terms are in the expansion of

(a+b)^6

Example 10

easy

Expand

(x+1)^2

Example 11

medium

Expand

(x+2)^3

Example 12

medium

Find the coefficient of

x^2

(2x+3)^4

Example 13

medium

Find the constant term in

\left(x+\frac{1}{x}\right)^4

Example 14

medium

What is the coefficient of

x^3

(1+x)^5

Example 15

medium

Expand

(2x-1)^3

Example 16

medium

Use the binomial theorem to compute

11^3

by writing

11=10+1

Example 17

medium

Find the middle term of

(a+b)^6

Example 18

medium

Why is

(a+b)^4 \ne a^4+b^4

? Give the missing terms.

Example 19

medium

Find the coefficient of

x^2

(1+x)^6

Example 20

challenge

Find the coefficient of

x^5

(2-x)^7

Example 21

challenge

Prove that the sum of the coefficients in

(a+b)^n

equals

2^n

Example 22

challenge

Find the term containing

x^4

\left(x^2+\frac{2}{x}\right)^5

Example 23

easy

Compute

\binom{7}{3}

Example 24

easy

Write the general term in the expansion of

(a+b)^n

Example 25

easy

Find the first three terms of

(1+x)^5

Example 26

easy

Compute

\binom{6}{3}

Example 27

medium

Expand

(x-2)^4

Example 28

medium

Find the coefficient of

x^3

(1+2x)^5

Example 29

medium

Find the coefficient of

x^2y^3

(x+y)^5

Example 30

medium

Find the constant term in

\left(2x-\frac{1}{x}\right)^6

Example 31

medium

Sum of coefficients of

(3x-2)^5

when

x=1

Example 32

medium

Expand

(1+x)^4

and use it to evaluate

(1.01)^4

to 4 decimal places.

Example 33

medium

Find the coefficient of

x^4

(2x^2+3)^4

Example 34

medium

Find the term independent of

x

\left(x^2+\frac{1}{x}\right)^6

Example 35

hard

Find the coefficient of

x^6

(1+x+x^2)^4

Example 36

hard

Use the binomial theorem to compute

99^3

Example 37

hard

(2x-3y)^4

, find the coefficient of

x^2 y^2

Example 38

hard

Find the largest coefficient in

(1+x)^8

Example 39

hard

Find the coefficient of

x^5

(1+x)^4(1-x)^3

Example 40

challenge

Prove the Vandermonde identity:

\binom{m+n}{r}=\sum_{k=0}^r \binom{m}{k}\binom{n}{r-k}

Example 41

challenge

Find the coefficient of

x^{10}

(1+x^2+x^3)^5

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Related Concepts

Binomial Coefficient Exponents Polynomial Multiplication

Background Knowledge

These ideas may be useful before you work through the harder examples.

binomial coefficientexponents