TITLE: How do you calculate combinations with fractions (eg, "$\frac13$ choose $2$")?
QUESTION [5 upvotes]: I was given "$\frac13$ choose $2$" and asked to compute. I checked the answer and it's $-\frac19$, but I have no clue how to arrive at this answer.
How do you deal with fractions in the combination formula?

REPLY [2 votes]: A generalisation of the binomial coefficient is given for real (or even complex) values $\alpha$ by
\begin{align*}
\binom{\alpha}{n}:=
\begin{cases}
\frac{\alpha(\alpha-1)\cdots(\alpha-n+1)}{n!}&n\geq 0\tag{1}\\
0&n<0
\end{cases}
\end{align*}
Using (1) we can write
\begin{align*}
\color{blue}{\binom{\frac{1}{3}}{2}}
=\frac{\frac{1}{3}\cdot\left(-\frac{2}{3}\right)}{2\cdot 1}
&\color{blue}{=-\frac{1}{9}}
\end{align*}
The formula (1) can be found for instance as formula (5.1) in Concrete Mathematics by R. L. Graham, D. Knuth and O. Patashnik.