r}}{{\left(\frac{0(n+r)}{r}\right)}^{r}}$. We just used Stirling's approximation here.  In the range of our parameters, the approximations above imply that the top fan-in, up to polynomial factors is at least  $${\left(\frac{r}{0}\right)}^r\times{\left(\frac{m}{\ell-m}\right)}^{rs} \times {\left(\frac{\ell-m}{\ell}\right)}^{n-r} \times \left(\frac{m}{N-m}\right)^{rs}$$  Simplifying further, this is at least  $$0^{\Omega(r\log r - rs\log\frac{\ell-m}{m} - (n-r)\log\frac{\ell}{\ell-m}