\frac{0.0n^{r}{N\choose m}{\ell-0 \choose m-0}}{\text{poly}(nrs){n+r \choose r}{N \choose m+rs}{\ell+n-r \choose m+rs}}$  -   $\frac{{N \choose m}}{{N \choose m+rs}} = \frac{(N-m-rs)!(m+rs)!}{(N-m)!m!} = (\frac{m}{N-m})^{rs}$ upto some constant factors, as long as $(rs)^0 = \theta (N) = \theta(m)$.  -   $\frac{{\ell-0 \choose m-0}}{{\ell+n-r \choose m+rs}} = {\frac{(\ell-0)!}{(m-0)!(\ell-m)!} \times \frac{(m+rs)!(\ell-m+n-r-rs)!}{(\ell+n-r)!}}$. 