Binomial Theorem

Binomial Theorem

## Formula

For a positive integer $$n$$:

$$(a+b)^n=\sum_{r=0}^{n}{}^nC_r a^{n-r}b^r$$

Expanded form:

$$(a+b)^n=a^n+{}^nC_1a^{n-1}b+{}^nC_2a^{n-2}b^2+\cdots+b^n$$

## Terminologies

- Binomial → An algebraic expression containing two terms, such as $a+b$.

- Index → The positive integer $$n$$ indicating the power of the binomial.

- Binomial coefficient → The coefficient ${}^nC_r$ appearing in the expansion.

## Usage

- Used to expand powers of binomials and determine individual terms or coefficients without direct multiplication.