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 _________.

A directed spanning tree (arborescence) rooted at vertex $5$ must satisfy the following properties:

1. The root vertex $5$ has an in-degree of $0$.

2. Every other vertex $j \in \{1, 2, 3, 4\}$ must have an in-degree of exactly $1$.

3. The subgraph contains no cycles (ignoring self-loops).

From the adjacency matrix definition, a directed edge exists from $i$ to $j$ ($i \to j$) if and only if $j \le i$. Ignoring self-loops ($i = j$), edges only go from a higher-numbered vertex to a strictly lower-numbered vertex ($i > j$). Therefore, the graph is a Directed Acyclic Graph (DAG), ensuring that any choice of incoming edges cannot form a cycle.

For each vertex $j \in \{1, 2, 3, 4\}$, we choose exactly one incoming edge from a vertex $i$ such that $i > j$:

• For vertex $4$: possible incoming edges from $\{5\} \implies 1$ choice.

• For vertex $3$: possible incoming edges from $\{4, 5\} \implies 2$ choices.

• For vertex $2$: possible incoming edges from $\{3, 4, 5\} \implies 3$ choices.

• For vertex $1$: possible incoming edges from $\{2, 3, 4, 5\} \implies 4$ choices.

The total number of directed spanning trees rooted at vertex $5$ is: