Simplification of factorial

Question

Can $\frac{(N_2-k)!}{(N_1-k)!}$ be expressed using $\Delta N = N_2-N_1$ as the only variable?

It stems from $\binom {N_2-k}{N_1-k}=\frac{(N_2-k)!}{(N_1-k)!(N_2-N_1)!}$.

Thanks!

Answer 1

No.

For $N_1=1$, $N_2=2$ and $k=1$ your xpression gives $$ \frac{(2-1)!}{(1-1)!}=\frac{1!}{0!}=\frac11=1 $$

For $N_1=3$, $N_2=4$ and $k=1$ it gives $$ \frac{(4-1)!}{(3-1)!}=\frac{3!}{2!}=\frac62=3 $$

In both cases $\Delta N=N_2-N_1=1$.