On some estimates for Erd\"os-R\`enyi random graph

We consider a number ${\nu }_n$ of components in a random graph $G(n,p)$ with $n$ vertices, where the probability of an edge is equal to $p$. By operating with special generating functions we shows the next asymptotic relation for factorial moments of ${\nu }_n$: $$\mathsf{E}({\nu }_n-1{)}^{\underset{‾}{s}}=(1+o(1)){(\frac{1}{p}\sum _{k=1}^{\mathrm{\infty }}\frac{k^{k-2}}{k!}(np{q}^n{)}^k)}^s+o(1)$$ as $n$ tends to $\mathrm{\infty }$ and $q=1-p$. And the following inequations hold: $$1-2n{q}^{n-1}\le p_n\le \frac{1}{n{q}^n},$$ $$1-\frac{1}{n{q}^n}\le p{i}_n\le n{q}^{n-1},$$ where $p_n$ is the probability that $G(n,p)$ is connected and $p{i}_n$ is the probability that $G(n,p)$ has an isolated vertex.