The binomial theorem is a formula for the expansion of $(a+b)^n$ , for $n$ a positive integer and $a$ and $b$ any two real (or complex) numbers, into a sum of powers of $a$ and $b$ . More precisely, $$(a+b)^n = a^n + \binom{n}{1} a^{n-1}b + \binom{n}{2} a^{n-2}b^2 + \cdots + b^n . $$ For example, if $n$ is 3 or 4, we have: \begin{eqnarray*} (a+b)^3 &= a^3 + 3 a^2 b + 3 a b^2 + b^3 \\ (a+b)^4 &= a^4 + 4 a^3 b + 6 a^2 b^2 + 4 a b^3 + b^4 . \end{eqnarray*} This result actually holds more generally if $a$ and $b$ belong to a commutative rig.