Simple probabilistic expression

Question

For the following expression:

$$ \prod_{i=0}^{n-1} \frac{2n-i}{3n-i} $$

I'm trying to get a simple expression, unsuccessfully.

Many thanks, Jonathan

Answer 1

$$\begin{align*} \prod_{i=0}^{n-1}\frac{2n-i}{3n-i}&=\frac{\frac{(2n)!}{n!}}{\frac{(3n)!}{(2n)!}}\\\\ &=\frac{\binom{2n}nn!}{\binom{3n}nn!}\\\\ &=\frac{\binom{2n}n}{\binom{3n}n}\;. \end{align*}$$

Of course you can replace the denominator with $\binom{3n}{2n}$ if you prefer. I doubt that you can get much nicer than this, though you can use Stirling’s approximation to get an estimate for large $n$.