I am working on extending the Erdös-Rényi paper "On the evolution of random graphs" (http://www.renyi.hu/~p_erdos/1960-10.pdf). In theorem 2a they calculate the number of isolated trees of order k.
I cannot understand equation (2.13):
$M(\epsilon(S))=\frac{{{n-k \choose 2} \choose N-k+1}}{{n \choose 2} \choose N}=(\frac{2N}{n^2})^{k-1}e^{\frac{-2Nk}{n}}(1+O(\frac{N}{n^2}))$.
In equation (7) they state that they often use:
${n \choose k}\sim\frac{{n^k}e^{-\frac{k^2}{2n}-\frac{k^3}{6n^2}}}{k!}$.
With this I can replicate the first part $(\frac{2N}{n^2})^{k-1}$.
Can someone explain me how to get to the second part $e^{\frac{-2Nk}{n}}$?