Binomial Theorem

combinatorics algebra

Definition

Binomial Theorem

For and ,

The -th term counts the ways to choose factors of (and factors of ) from the -fold product , weighted by the binomial coefficient .

Special Cases

The endpoints of the sum recover two basic identities:

and the middle term at gives the familiar , with the coefficient .

Generalisation

The theorem extends to non-integer or negative exponents via the generalised binomial series. For any real (or complex) and ,

The finite sum above is the instance of this series, where for truncates the sum.