# Article

# Article

- Mathematics
- Algebra and number theory
- Binomial theorem

- Mathematics
- Probability, statistics, combinatorial theory
- Binomial theorem

# Binomial theorem

Article By:

**Cooley, Hollis R. **Formerly, Department of Mathematics, New York University, New York, New York.

Last reviewed:2014

DOI:https://doi.org/10.1036/1097-8542.081600

**One of the most important algebraic identities, with many applications in a variety of fields.** The binomial theorem, discovered by Isaac Newton, expresses the result of multiplying a binomial by itself any number of times, as given in Eq. (1),

*c*

_{1},

*c*

_{2},

*c*

_{3}, … ,

*c*

_{r}, … , are given by Eqs. (2).

*b*

^{r}in the expansion of (

*a*+

*b*)

^{n}. It is a fraction with the numerator and denominator each containing

*r*factors; those in the denominator begin with 1 and increase by 1; those in the numerator begin with

*n*and decrease by 1. It is easily shown that Eq. (4)

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