Binomial Theorem

combinatorics algebra

Definition

Binomial Theorem

For $n\in {\mathbb{Z}}_{\ge 0}$ and $x,y\in \mathbb{C}$,

$$(x+y{)}^n=\sum _{k=0}^n(\genfrac{}{}{0ex}{}{n}{k})x^{n-k}{y}^k.$$

The $k$-th term counts the ways to choose $k$ factors of $y$ (and $n-k$ factors of $x$) from the $n$-fold product $(x+y)(x+y)\cdots (x+y)$, weighted by the binomial coefficient $\left(\genfrac{}{}{0ex}{}{n}{k}\right)$.

Special Cases

The endpoints of the sum recover two basic identities:

$$k=0:\left(\genfrac{}{}{0ex}{}{n}{0}\right)x^n{y}^0=x^n,k=n:\left(\genfrac{}{}{0ex}{}{n}{n}\right)x^0{y}^n=y^n,$$

and the middle term at $n=2$ gives the familiar $(x+y{)}^2=x^2+2xy+y^2$, with the coefficient $2=\left(\genfrac{}{}{0ex}{}{2}{1}\right)$.

Generalisation

The theorem extends to non-integer or negative exponents via the generalised binomial series. For any real (or complex) $\alpha$ and $|x|<1$,

$$(1+x{)}^{\alpha }=\sum _{k=0}^{\infty }(\genfrac{}{}{0ex}{}{\alpha }{k})x^k,(\genfrac{}{}{0ex}{}{\alpha }{k}):=\frac{\alpha (\alpha -1)\cdots (\alpha -k+1)}{k!}.$$

The finite sum above is the $\alpha =n\in {\mathbb{Z}}_{\ge 0}$ instance of this series, where $\left(\genfrac{}{}{0ex}{}{n}{k}\right)=0$ for $k>n$ truncates the sum.