Let $G(V,E)$ be a directed graph, where $V = \{1,2,3,4,5\}$ is the set of vertices and $E$ is the set of directed edges, as defined by the following adjacency matrix $A$. $$A[i][j] = \begin{cases} 1, & 1 \le j \le i \le 5 \\ 0, & \text{otherwise} \end{cases}$$ $A[i][j] = 1$ indicates a directed edge from node $i$ to node $j$. A *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 _________.

Let G(V,E) be a directed graph, where V = \1,2,3,4,5\ is the set of vertices and E is the set of directed edges, as defined by the following adjacency matrix A. A[i][j] = cases 1, & 1 j i 5 \\ 0, & otherwise cases A[i][j] = 1 indicates a directed edge from node i to node j. A *directed spanning tree* of G, rooted at r 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 …

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