Find total count of all paths starting from a fixed vertex to all other vertexes of the graph
Suppose you have a polynomial-time algorithm $\mathcal{A}$ to solve this problem, then consider a graph $G$ and arbitrary two vertices $s$ and $t$. We construct a new graph $G'$ by add a new vertex $t'$ into $G$, as well as an edge $(t,t')$. Now we have

\begin{align}
\text{#(paths from $s$ to $t$ in $G$)} =&amp;\ \text{#(paths from $s$ to all other vertices in $G'$)}\\
&amp;-\text{#(paths from $s$ to all other vertices in $G$)}.
\end{align}

That means, we can use $\mathcal{A}$ to count the number of paths between two vertices in polynomial time. However, the latter problem is #P-hard. So such an algorithm $\mathcal{A}$ does not exist unless $\mathbf{P}=\mathbf{NP}$.