If $N_c=\lfloor \frac{1}{2}n\log n+cn\rfloor$ for some integer $n$ and real constant $c$, then how would one go about showing the following identity where $k$ is a fixed integer:
$\lim_{n\rightarrow \infty} \binom{n}{k}\frac{\binom{\binom{n-k}{2}}{N_c}}{\binom{\binom{n}{2}}{N_c}}=\frac{e^{-2kc}}{k!}.$
It feels like using Stirling's approximation would help but I can't quite figure out how...
I ask this question because I am currently trying to understand the paper in which Erdős initiated the study of the evolution of random graphs
P. Erdős and A. Rényi, On random graphs I, Publicationes Mathematicae Debrecen 6 (1959), pp.290–297, Erdős Project link.
where understanding this quantity is a key step in computing the probability that a graph on $n$ vertices with $N_c$ randomly chosen edges is completely connected.