The Binomial Theorem

The binomial theorem gives a systematic way to expand a power of a two-term sum, $(a+b)^n$, without multiplying the bracket out repeatedly. For a positive integer $n$, $$(a+b)^n = \sum_{r=0}^{n} \binom{n}{r} a^{n-r} b^{r} = \binom{n}{0}a^n + \binom{n}{1}a^{n-1}b + \binom{n}{2}a^{n-2}b^2 + \cdots + \binom{n}{n}b^n.$$ Notice the structure of every expansion: there are $n+1$ terms; the power of $a$ starts at $n$ and falls to 0 while the power of $b$ rises from 0 to $n$; and in every term the two exponents add to $n$. The numbers $\binom{n}{r}$ multiplying each term are the binomial coefficients.

A binomial coefficient $\binom{n}{r}$ (read "$n$ choose $r$") counts the number of ways to choose $r$ objects from $n$, and is computed by $$\binom{n}{r} = \frac{n!}{r!\,(n-r)!},$$ where $n! = n(n-1)(