Q48

Let $\textit{G(V,E)}$ be a directed graph, where $\textit{V} = \{ 1, 2, 3, 4, 5 \}$ is the set of vertices and $\textit{E}$ is the set of directed edges, as defined by the following adjacency matrix $\textit{A}.$

$$A [i][j] = \left\{\begin{matrix} 1, & 1 \leq j \leq i \leq 5 \\ 0, & \text{otherwise}\end{matrix}\right.$$

$A[i][j] = 1$ indicates a directed edge from node $i$ to node $j.$ A $\textit{directed spanning tree}$ of $G,$ rooted at $r \in V,$ is defined as a subgraph $T$ of $G$ such that the undirected version of $T$ is a tree, and $T$ contains a directed path from $r$ to every other vertex in $V.$ The number of such directed spanning trees rooted at vertex $5$ is __________________.