I have a problem with evaluating a probability that random binomial graph $\mathbb{G}_{n,p}$ contains fixed path made from edges $e_1$, $e_2$, $\ldots$, $e_n$, $1\leq l \leq n$. What is important, the edges in path can not repeat.
My attempt.
Let $m$ be an amount of edges and $n$ an amount of vertices. Let's assume that we already have $l$ edges fixed.
$\mathbb{P}(l=1)=p^{m-1}(1-p)^{{{n}\choose{2}}- (m-1)}$,
$\mathbb{P}(l=2)=p^{m-2}(1-p)^{{{n}\choose{2}}- (m-2)}$,
$\ldots$,
$\mathbb{P}(l=2)=p^{m-n}(1-p)^{{{n}\choose{2}}- (m-n)}$.
So the probability is $\sum_{i=1}^l p^{m-i}(1-p)^{{{n}\choose{2}}- (m-i)}$. Is this reasoning correct? If not, I would really appreciate showing me correct solution.