I'm in the process of working through Erdős and Rényi's 1959 article "On Random Graphs I". In the proof of the first Lemma, equation 14 gives a bound on an expression involving several binomial coefficients:
$$\binom{n}{s}\frac{\displaystyle\binom{\binom{n}{2}-s(n-s)}{N_c}}{\displaystyle\binom{\binom{n}{2}}{N_c}}\le\frac{e^{(3-2c)s}}{s!}$$
where $N_c = [\frac{1}{2}n\log{n} + c n]$, $s\le n/2$ and $n > n_0$ for some $n_0$, and $[x]$ denotes the largest integer less than or equal to $x$. (Equation 15 gives a corresponding expression for the case $s \ge n/2$.)
After spending some time playing around with Stirling's approximation and bounds such as $\left(\frac{n}{k}\right)^k \le \binom{n}{k} \le \frac{n^k}{k!} \le \left(\frac{n \cdot e}{k}\right)^k$, I don't feel that I'm any closer to understanding where this bound comes from.
It seems like there ought to be a fairly straightforward procedure for obtaining such bounds, given that the authors didn't feel any need to elaborate. I'd be very grateful for any pointers on how to proceed!