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Binomial coefficients are the coefficients in the expanded version of a binomial, such as \((x+y)^5\text{.}\) What happens when we multiply such a binomial out? We will expand \((x+y)^n\) for various values of \(n\text{.}\) Each of these are done by multiplying everything out (i.e., FOIL-ing) and then collecting like terms.

\begin{equation*} (x+y)^1 = x + y \end{equation*}

\begin{equation*} (x+y)^2 = x^2 + 2xy + y^2 \end{equation*}

\begin{equation*} (x+y)^3 = x^3 + 3x^2y + 3xy^2 + y^3 \end{equation*}

\begin{equation*} (x+y)^4 = x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4\text{.} \end{equation*}

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