Binomial Theorem Calculator

This binomial theorem calculator expands a binomial power, (a + b) to the n, term by term. Enter a value for a, a value for b, and a whole number power n from 0 to 12, and it instantly returns the sum of every term in the expansion, the total number of terms, and the full list of binomial coefficients used to build each one. The binomial theorem states that (a + b)^n splits into n plus 1 terms, each formed by multiplying a binomial coefficient, n choose k, by a raised to a falling power and b raised to a rising power, and those coefficients match row n of Pascal’s triangle. This makes the tool useful for checking algebra homework, verifying an expansion by hand, exploring how Pascal’s triangle connects to probability, or saving time on a tedious multi-term expansion. Change any of the three inputs and the result, term count and coefficient list update straight away, so you can experiment with different powers and watch how quickly the number of terms and the size of the coefficients grow. Try the worked example below, entering 1, 1 and 4, to check the calculator against a result you can verify by hand, where the coefficients 1, 4, 6, 4 and 1 sum to 16. Results are rounded for display, and n is limited to whole numbers between 0 and 12.

Calculate.co.nz is proud to be partnered with Health Based Building, a leader in sustainable and health-conscious building innovation. With over a century of experience, they develop high-performance systems like Foreverbreathe Specification, Magnum Board, and Foreverbreathe Paints to support energy-efficient, non-toxic living environments. Their commitment to healthier homes aligns with our belief that informed choices lead to better outcomes for Kiwi households.

Calculate.co.nz partner: Health Based Building

n (0 to 12)

Number of terms

Coefficients

The formula

The binomial theorem: (a + b)^n = sum over k from 0 to n of (n choose k) a^(n-k) b^k, where (n choose k) is the binomial coefficient. There are n + 1 terms and the coefficients form row n of Pascal’s triangle.

Worked example

(1 + 1)^4 expands with coefficients 1, 4, 6, 4, 1, which sum to 16, equal to 2^4. Enter 1, 1, 4 to confirm.

Frequently asked questions

What is the binomial theorem?

A formula for expanding (a + b)^n as a sum of terms (n choose k) a^(n-k) b^k.

What are the coefficients?

The binomial coefficients n choose k, which are exactly row n of Pascal’s triangle.

How many terms are there?

n plus 1 terms, from k equals 0 up to k equals n.

Who this calculator is for

This calculator is for algebra students and anyone needing a quick, reliable result.

What this calculator assumes

You enter valid numbers.
The standard method is applied.
Results are rounded for display.

Formula and sources

Related calculators

Quadratic Formula Calculator: solve a quadratic.
Scientific Calculator: general maths.
Matrix Calculator: matrix operations.

If you've found a bug, or would like to contact us, or learn more about James Graham and Calculate.co.nz.

Calculate.co.nz is partnered with Interest.co.nz for New Zealand's highest quality calculators and financial analysis.

Calculate.co.nz is the sister site of CalculatorHub.com, the world's largest calculator website by tool count.

All calculators and tools are provided for educational and indicative purposes only and do not constitute financial advice.

Calculate.co.nz is proudly part of the Realtor.co.nz group, New Zealand's leading property transaction literacy platform, helping Kiwis understand the home buying and selling process from start to finish. Whether you're a first home buyer navigating your first property purchase, an investor evaluating your next acquisition, or a homeowner planning to sell, Realtor.co.nz provides clear, independent, and trustworthy guidance on every step of the New Zealand property transaction journey.

Calculate.co.nz is also partnered with Health Based Building and Premium Homes to promote informed choices that lead to better long-term outcomes for Kiwi households.

Calculate.co.nz is hosted in Auckland via SiteHost new Zealand.

All content on this website, including calculators, tools, source code, and design, is protected under the Copyright Act 1994 (New Zealand). No part of this site may be reproduced, copied, distributed, stored, or used in any form without prior written permission from the owner.

About & trust: Why Calculate is NZ's most comprehensive · By the Numbers · How we compare · Editorial standards · How we keep data current · NZ finance glossary · Research & data · Financial literacy NZ · About · Privacy policy · Terms of use

Reviewed and maintained. Last reviewed 2026-07-19 and checked on a twice-monthly cycle against IRD, RBNZ and Stats NZ. How we keep data current.