(i) Define the two Erd ̋os–R ́enyi models, $G(n,m)$ and $G(n,p)$, of random graphs.
(ii) For a fixed value of $n$, describe each of the two models as a probability distribution on the set $G_n$ of all graphs on the $n$ nodes $X=\{1,2,...,n\}$.
(iii) For $n=100$ and $p=\frac{1}{99}$, what is the probability that a graph G sampled from the $G(n,p)$ model has exactly 50 edges?
For (iii) what I've done is:
$$P = \binom{\binom{100}{2}}{50}*\left(\frac{1}{99}\right)^{50}*\left(\frac{98}{99}\right)^{\binom{100}{2}-50}.$$
Is my thinking correct? I know the expected edges is 50 via the formula.