Unit-I: Sets and Functions
Unit-II: Algebra
Unit-III: Coordinate Geometry
Unit-IV: Calculus
Unit-V: Mathematical Reasoning
Unit-VI: Statistics and Probability

Statement of Binomial Theorem

It is possible to expand any non-negative power of  x+y is the sum of the form, i.e.,

 (x+y)^n=^nC_0x^ny^0+^nC_1x^{n-1}y^1+...+^nC_rx^{n-r}y^r+...+^nC_{n-1}x^1y^{n-1}+^nC_nx^0y^n

where 0 \leq r \leq n , and each  ^nC_k is a positive integer known as Binomial Co-efficient.

This formula is also referred to as the Binomial Formula or Binomial Identity.

Using Summation Notation, it can be written as

 (x+y)^n=\sum_{k=0}^{n} ^nC_k x^{n-k}y^k=\sum_{k=0}^{n} ^nC_k x^ky^{n-k}

 \begin{aligned}\longrightarrow(x+y)^n&=^nC_0x^ny^0+^nC_1x^{n-1}y^1+...+^nC_{n-1}x^1y^{n-1}+^nC_nx^0y^n\\&=x^n+^nC_1x^{n-1}y+...+ ^nC_{n-1}xy^{n-1}+y^n\\&=x^n+nx^{n-1}y+...+nxy^{n-1}+y^n\end{aligned}

\begin{aligned}\text { For }n=2,(x+y)^2&=2C_0x^2y^0+^2C_1x^1y^1+^2C_2x^0y^2\\&=x^2+2xy+y^2\end{aligned}

Proving the above formula by Binomial Theorem.

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