Binomial coefficients- how to work out 0.5 choose 2

Question

I was recently given a question in my homework, and part of the formula asks for 0.5 choose 2, how would I work out the decimal factorial?

0.5 c 2

Answer 1

$$\binom{n}{r}=\frac{n!}{r!(n-r)!}$$ However as $n\notin\Bbb Z$, the typical definition of $n!=1\cdot2\cdots n$ is not applicable. Instead we use the falling factorial $n^{\underline{r}}=n(n-1)\cdots(n-(r-1))$ and the $(n-r)!$ vanishes (To see why, try integer $n$ and $r$ using the falling factorials, and you'll see that $n^{\underline{r}}=\frac{n!}{(n-r)!}$).

So we get: $$\binom{n}{r}=\frac{n^{\underline{r}}}{r!}$$ In the case of $n=\frac 12, r=2$, we get $$\binom{\frac 12}{2}=\frac{(\frac12)(\frac12-1)}{2!}=\frac{-\frac14}{2}=-\frac18$$