Binomial Theorem

The expansion of power of a binomial as sum of the terms is called the binomial theorem.

$(x+y)^{\displaystyle n}$ $\,=\,$ $\displaystyle \binom{n}{0}\,x^{\Large n}\,y^{\large 0}$ $+$ $\displaystyle \binom{n}{1}\,x^{\Large n \large \,-\,1}\,y^{\large 1}$ $+$ $\displaystyle \binom{n}{2}\,x^{\Large n \large \,-\,2}\,y^{\large 2}$ $+$ $\displaystyle \binom{n}{3}\,x^{\Large n \large \,-\,3}\,y^{\large 3}$ $+$ $\cdots$ $+$ $\displaystyle \binom{n}{n}\,x^{\large 0}\,y^{\Large n}$

In mathematics, the binomial theorem is actually written in algebraic form and it is also written in the following mathematical form.

$(x+y)^{\displaystyle n}$ $\,=\,$ $^{\displaystyle n}C_{\large 0}\,x^{\Large n}\,y^{\large 0}$ $+$ $^{\displaystyle n}C_{\large 1}\,x^{\Large n \large \,-\,1}\,y^{\large 1}$ $+$ $^{\displaystyle n}C_{\large 2}\,x^{\Large n \large \,-\,2}\,y^{\large 2}$ $+$ $^{\displaystyle n}C_{\large 3}\,x^{\Large n \large \,-\,3}\,y^{\large 3}$ $+$ $\cdots$ $+$ $^{\displaystyle n}C_{\displaystyle n}\,x^{\large 0}\,y^{\Large n}$

In mathematics, you can express the binomial theorem in any one of the above two methods. Now, let’s learn the binomial theorem in detail.

Introduction

Let $x$ and $y$ be two variables and they belong to real numbers. It is mathematically written as $x, y ∈ R$. $n$ is a constant and it belongs to positive integers. It is written as $n ∈ I^+$. The sum of the variables $x$ and $y$ forms a binomial $x+y$. The $n^{th}$ power of binomial is written as $(x+y)^{\displaystyle n}$ in mathematics.

The $n^{th}$ power of binomial $x+y$

$(1).\,\,\,$ $(x+y)^{\displaystyle n}$ $\,=\,$ $\displaystyle \binom{n}{0}\,x^{\Large n}\,y^{\large 0}$ $+$ $\displaystyle \binom{n}{1}\,x^{\Large n \large \,-\,1}\,y^{\large 1}$ $+$ $\displaystyle \binom{n}{2}\,x^{\Large n \large \,-\,2}\,y^{\large 2}$ $+$ $\displaystyle \binom{n}{3}\,x^{\Large n \large \,-\,3}\,y^{\large 3}$ $+$ $\cdots$ $+$ $\displaystyle \binom{n}{n}\,x^{\large 0}\,y^{\Large n}$

$(2).\,\,\,$ $(x+y)^{\displaystyle n}$ $\,=\,$ $^{\displaystyle n}C_{\large 0}\,x^{\Large n}\,y^{\large 0}$ $+$ $^{\displaystyle n}C_{\large 1}\,x^{\Large n \large \,-\,1}\,y^{\large 1}$ $+$ $^{\displaystyle n}C_{\large 2}\,x^{\Large n \large \,-\,2}\,y^{\large 2}$ $+$ $^{\displaystyle n}C_{\large 3}\,x^{\Large n \large \,-\,3}\,y^{\large 3}$ $+$ $\cdots$ $+$ $^{\displaystyle n}C_{\displaystyle n}\,x^{\large 0}\,y^{\Large n}$

Math Questions

The math problems with solutions to learn how to solve a problem.

Learn solutions

Practice now

Math Videos

The math videos tutorials with visual graphics to learn every concept.

Watch now

Subscribe us

Get the latest math updates from the Math Doubts by subscribing us.