\times \frac{(N-(n-d-r))!}{N!}\times \frac{m!}{(m-(n-d-r))!} \leq 0$$  Now, using the approximation from Lemma \[lem:approx\], $$\begin{aligned} \log\frac{N!}{(N-(n-d-r))!} &=& (n-d-r)\log N \pm O\left(\frac{(n-d-r)^0}{N}\right) \text{and} \\ \log\frac{m!}{(m-(n-d-r))!} &=& (n-d-r)\log m \pm O\left(\frac{(n-d-r)^0}{m}\right)\end{aligned}$$  