I am working through the paper On Random Graphs I by Erdős and Rényi (which can be found here), and I am stuck on deducing one of the particular inequalities used.
Here we are considering the set of random graphs with $n$ fixed vertices and $N$ edges. Let $N_c := \lceil \frac{1}{2} n \log n + cn \rceil$ edges, where $c$ is an arbitrary fixed real number.
My question is about the second inequality in
$$ P(\overline{A} E_{\,\log\log n}, n, N_c) \leq \sum^{\log \log n}_{s=2} \binom{n}{s} \left\{ \sum^{\binom{s}{2}}_{r=1} \binom{\binom{s}{2}}{r} \frac{\begin{pmatrix} \binom{n-s}{2} \\ N_c - r \end{pmatrix}}{\begin{pmatrix} \binom{n}{2} \\ N_c \end{pmatrix}} \right\} \leq \frac{\log n}{n} \sum^{\log \log n}_{s=2} \frac{2^{\binom{s}{2} e^{-2sc}}}{s!} . $$ (This is a bound for the probability that a random graph is in $\overline{A}$ (not of "Type A", where a Type A graph is one such that all the vertices outside of the greatest connected component of the graph are isolated) and in $E_{\, \log \log n}$ (the number of vertices in the greatest connected component have at least $n - \log \log n$ points).)
The first inequality (19) arises from a counting argument, but I am unsure about how exactly the second inequality (20) can be deduced.
I am thinking that the property $ \sum^{\binom{s}{2}}_{r=1} \begin{pmatrix} \binom{s}{2} \\ r \end{pmatrix} \leq 2^{\binom{s}{2}} $ will be useful, giving us one of the desired terms. Then if I can somehow obtain a bound for the remaining terms in the inner sum $\binom{n}{s} \cdot \begin{pmatrix} \binom{n-s}{2} \\ N_c - r \end{pmatrix} \Big/ \begin{pmatrix} \binom{n}{2} \\ N_c \end{pmatrix}$ which is independent of $r$ and looks like $\frac{\log n}{n} \frac{e^{-2sc}}{s!}$, this would imply the desired inequality. However, I am not too sure how this can be done.