## Section2.6The Binomial Theorem

Here is a truly basic result from combinatorics kindergarten.

View $$(x+y)^n$$ as a product

\begin{equation*} (x+y)^n=\underbrace{(x+y)(x+y)(x+y)(x+y)\dots(x+y)(x+y)}_{n\text{ factors} }. \end{equation*}

Each term of the expansion of the product results from choosing either $$x$$ or $$y$$ from one of these factors. If $$x$$ is chosen $$n-i$$ times and $$y$$ is chosen $$i$$ times, then the resulting product is $$x^{n-i}y^i\text{.}$$ Clearly, the number of such terms is $$C(n,i)\text{,}$$ i.e., out of the $$n$$ factors, we choose the element $$y$$ from $$i$$ of them, while we take $$x$$ in the remaining $$n-i\text{.}$$

### Example2.31.

There are times when we are interested not in the full expansion of a power of a binomial, but just the coefficient on one of the terms. The Binomial Theorem gives that the coefficient of $$x^5y^8$$ in $$(2x-3y)^{13}$$ is $$\binom{13}{5}2^{5}(-3)^8\text{.}$$