Is the following equation valid for a binomial coefficient?

Question

From my notes on the binomial series I deduced a formula for the value of a binomial equation using product notation:

$ {n \choose r} = {\frac {\prod_{\lambda=0}^r(n-\lambda)} {r!}}$

I believe this holds true for $ n \in \mathbb R, r \in \mathbb Z^+$ but I have not seen it in any publications. Is this a general formula used?

Answer 1

We can find a corresponding formula of the binomial coefficients even as definition in section 5.1 Binomial Coefficients, formula (5.1) in Concrete Mathematics by R.L. Graham, D.E. Knuth and O. Patashnik. \begin{align*} \binom{n}{r}:=\frac{\prod_{\lambda=0}^{r-1}(n-\lambda)}{r!}\qquad n\in\mathbb{R}, r\in\mathbb{Z}^{+}\\ \end{align*}

Note the upper limit of the product is $r-1$.