Binomial Theorem

Pronunciation: /baɪˈnoʊ mi əl ˈθiər əm/ ?

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.[1] 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. binomial theorem. Encyclopedia Britannica. (Accessed: 2009-03-12). theorem.
  2. 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. (Accessed: 2009-12-19).

More Information

  • McAdams, David. Binomial. All Math Words Encyclopedia. Life is a Story Problem LLC. 2009-03-12.

Cite this article as:

Binomial Theorem. 2009-12-19. All Math Words Encyclopedia. Life is a Story Problem LLC.


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Revision History

2009-12-19: Added "References" (McAdams, David.)
2008-12-03: Changed equations to images (McAdams, David.)
2008-06-28: Replaced equations with dHot_Eqn.class (McAdams, David.)
2008-06-07: Corrected link errors. Corrected spelling (McAdams, David.)
2008-04-18: Initial version (McAdams, David.)

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