Binomial theorem

The binomial theorem, for $n\in \mathbb{N}$ is given by:

$$(a+b{)}^n=a^n+(\genfrac{}{}{0ex}{}{n}{1})a^{n-1}b+\cdots +(\genfrac{}{}{0ex}{}{n}{r})a^{n-r}{b}^r+\cdots +b^n$$

Note the use of combinations.

Series

We can further generalise the binomial theorem with the binomial series, for when $p\notin \mathbb{Z}$. It’s the Maclaurin expansion for $f(x)=(1+x{)}^p$.

$$1+px+\frac{p(p-1)}{2!}{x}^2+\frac{p(p-1)(p-2)}{3!}{x}^3+\cdots +\frac{p(p-1)\dots (p-n+1)}{n!}{x}^n+\dots$$

Probability

see binomial probability law $a=p$, $b=p-1$