Let $A$ be a local ring with maximal ideal $m$ that is $m$-adically complete, and assume $1/2 \in A^\times$. I've read in several places that for any $x \in m$, a square root of $1 + x$ in $A$ is given by the binomial series
$ \sum_{n = 0}^\infty {{1/2} \choose n} x^n, $ where ${{1/2} \choose n} = \frac{(1/2)(1/2 - 1) \cdots (1/2 - n + 1)}{n!}$. I don't understand why these binomial coefficients make sense in $A$. We have $n!$ in the denominator, so why is $n! \in A^\times$?