Product of binomial coefficients

Question

Is there any way to simplify given expression ($j$ and $i$ are given, $n\leq \lfloor j/i \rfloor$) $$\prod_{x=1}^n \binom {j-(x-1)i} {i}$$

(e.g. in terms of factorials)?

Thanks!

Answer 1

\begin{align*} \prod_{x=1}^n \binom{j-(x-1)i}{i} &= \binom{j}{i}\binom{j-i}{i}\binom{j-2i}{i}\cdots\binom{j-(n-1)i}{i} \\ &= \frac{j!}{(j-i)!i!} \frac{(j-i)!}{(j-2i)!i!}\frac{(j-2i)!}{(j-3i)!i!} \cdots \frac{(j-(n-1)i)!}{(j-ni)!i!} \\ &= \frac{j!}{(j-ni)! (i!)^n} = \binom{j}{j-ni \quad i \quad i \quad \cdots \quad i} \end{align*} is the only thing I can think of. Hope it helps...