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 $x+y$

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

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

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