A generalization of formulae such as ${\left(x+y\right)}^2=x^2+2xy+y^2$ and ${\left(x+y\right)}^3=x^3+3x^{2}y+3x{y}^2+y^3$ used in elementary algebra. The Binomial Theorem states: for all positive integers n,

where $\left(\begin{array}{c}n\\ r\end{array}\right)=\frac{n!}{r!\left(n-r\right)!}$ (see binomial coefficient).

When y = 1, the theorem agrees with the terminating binomial series. There is a multinomial theorem for more general expansions such as (x₁ + x₂ + ··x_k)ⁿ.