Binomial Theorem

Pronunciation: /baɪˈnoʊ.mi.əl ˈθiər.əm/ Explain

The binomial theorem states that when raising a binomial to an integer power, the binomial coefficients can be calculated using the combination

choose(n,m) = (n!)/((n-k)!k!)
where n is the integer power, and k is the order of the coefficient.[2] This is expressed mathematically with the expression
n=sum from k=0 to n (choose(n,k)x^(n-k)a^(k))
Start with the simple binomial
In this expression
So the first term is
The next two terms are
This gives a result of
You can use long multiplication to verify this result.


  1. McAdams, David E.. All Math Words Dictionary, binomial theorem. 2nd Classroom edition 20150108-4799968. pg 27. Life is a Story Problem LLC. January 8, 2015. Buy the book
  2. binomial theorem. Encyclopedia Britannica. Merriam-Webster. Last Accessed 6/19/2018. theorem. Buy the book
  3. Fine, Henry B., Ph. D.. Number-System of Algebra Treated Theoretically and Historically. 2nd edition. pp 44-53. D. C. Heath & Co., Boston, U.S.A.. 1907. Last Accessed 6/19/2018. Buy the book

More Information

  • McAdams, David E.. Binomial. All Math Words Encyclopedia. Life is a Story Problem LLC. 6/22/2018.

Cite this article as:

McAdams, David E. Binomial Theorem. 12/21/2018. All Math Words Encyclopedia. Life is a Story Problem LLC.

Image Credits

Revision History

12/21/2018: Reviewed and corrected IPA pronunication. (McAdams, David E.)
6/22/2018: Removed broken links, updated license, implemented new markup. (McAdams, David E.)
12/19/2009: Added "References". (McAdams, David E.)
12/3/2008: Changed equations to images. (McAdams, David E.)
6/28/2008: Replaced equations with dHot_Eqn.class (McAdams, David E.)
6/7/2008: Corrected link errors. Corrected spelling (McAdams, David E.)
4/18/2008: Initial version. (McAdams, David E.)

All Math Words Encyclopedia is a service of Life is a Story Problem LLC.
Copyright © 2018 Life is a Story Problem LLC. All rights reserved.
This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License