For a vertex with degree $k$, what is the probability that its clustering coefficient equals $c$ in an Erdős–Rényi graph?

Question

I'm making an assignment for one of my university courses, but one of the questions has completely stumped me. I know that the probability distribution for the degree of a single vertex is

$$P(k) = \binom{N - 1}{k}p^k(1-p)^{N-1-k}$$

but I have no idea how I can use this to solve the question in the title.

Answer 1

As Misha's comment states, there are $\binom{k}{2}$ potential edges among the vertex's neighbors. For each such edge, the probability for it to be in the random graph is $p$. Thus the total # of edges, denoted as $X$, among neighbors follows the binomial distribution; that is, $\mathsf{binomial}(\binom{k}{2}, p)$. Hence we have $$ \Pr\left(\frac{X}{\binom{k}{2}}=c\right) = \Pr\left(X = c\binom{k}{2}\right) = \binom{\binom{k}{2}}{c\binom{k}{2}}\cdot p^{c\binom{k}{2}}\cdot(1-p)^{\binom{k}{2} - c\binom{k}{2}} $$