GATE CSE 2022 Q58: Directed spanning trees rooted at vertex 5

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_____________.