Practice Binomial Theorem in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick 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.

Showing a random 20 of 50 problems.

Example 1

medium

Expand

(x + 2)^3

using the Binomial Theorem.

Example 2

easy

Write the coefficients in the expansion of

(a+b)^4

Example 3

medium

Sum of coefficients of

(3x-2)^5

when

x=1

Example 4

medium

Use the binomial theorem to compute

11^3

by writing

11=10+1

Example 5

challenge

Prove that the sum of the coefficients in

(a+b)^n

equals

2^n

Example 6

hard

Find the coefficient of

x^6

(1+x+x^2)^4

Example 7

challenge

Prove the Vandermonde identity:

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

Example 8

medium

Find the coefficient of

x^4

(2x^2+3)^4

Example 9

medium

What is

\sum_{k=0}^n \binom{n}{k}=?

Example 10

hard

Find the coefficient of

x^3

in the expansion of

(2x + 3)^5

Example 11

hard

(2x-3y)^4

, find the coefficient of

x^2 y^2

Example 12

challenge

Find the term containing

x^4

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

Example 13

medium

Find the coefficient of

x^2

(2x+3)^4

Example 14

easy

Compute

\binom{5}{2}

Example 15

medium

Find the coefficient of

x^2y^3

(x+y)^5

Example 16

easy

Compute

\binom{7}{3}

Example 17

easy

Compute

\binom{6}{3}

Example 18

easy

Compute the binomial coefficient

\binom{4}{2}

Example 19

easy

What is the coefficient of

a^2b

in the expansion of

(a+b)^3

Example 20

medium

Expand

(1+x)^4

and use it to evaluate

(1.01)^4

to 4 decimal places.

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Related Concepts

Binomial Coefficient Exponents Polynomial Multiplication