# Binomial Theorem **A description of the algebraic expansion of powers of a binomial.** > [!Example] Binomial Theorem > The *binomial theorem* states that the polynomial $(x+y)^n$ expands into a sum involving terms of *only the form* $ax^{b}y^{c}$, where the exponents are nonnegative and $b+c=n$. > $\begin{align} > (x+y)^{n}&=\binom{n}{0}x^{n}y^{0}+\binom{n}{1}x^{n-1}y^{1}+\cdots+\binom{n}{n-1}x^{1}y^{n-1}+\binom{n}{n}x^{0}y^{n} \\ > &=\sum_{k=0}^{n}\binom{n}{k}x^{k}y^{n-k} > \end{align}$ > > The coefficient $a$ which is equal to $\binom{n}{b}$ or $\binom{n}{c}$ is the *binomial coefficient*. > [!Geometric explanation]- > A geometric explanation of the binomial theorem is possible for positive values of $x$ and $y$. > > By considering them as lengths, it becomes evident that geometric shapes representing the original polynomial can be divided into sections that correspond to all of the terms in the expansion. > > ![[BinomialTheoremGeometric.svg]] > > This is visually similar to the geometric proof of the [[power rule]]. ## Binomial coefficient The *binomial coefficient* can be computed using the formula $\binom{n}{k}=\frac{n!}{k!(n-k)!}$ Arranging the binomial coefficients $\binom{n}{0},\dots,\binom{n}{n}$ into successive rows corresponding to $n=0,1,2,\dots$ forms [[Pascal's triangle]]. Additionally, the binomial coefficient $\binom{n}{k}$ corresponds to the total number of different [[Combination|combinations]] of $n$ elements chosen from $k$ options.