1. **Problem Statement:** We have a graph with nodes $a$, $b$, and $c$ and edges: - $e_1$: loop on $a$ - $e_2$, $e_3$, $e_4$: three edges from $a$ to $b$ - $e_5$: edge from $b$ to $c$ We want to find: a. Number of paths from $a$ to $c$ b. Number of trails from $a$ to $c$ c. Number of walks from $a$ to $c$ --- 2. **Definitions and Rules:** - A **path** is a walk with no repeated vertices. - A **trail** is a walk with no repeated edges. - A **walk** can repeat vertices and edges any number of times. --- 3. **Counting Paths from $a$ to $c$:** - Since paths cannot repeat vertices, and $a$ and $c$ are distinct, the path must be $a \to b \to c$. - From $a$ to $b$, there are 3 edges ($e_2$, $e_3$, $e_4$), but since paths cannot repeat vertices, only one edge can be used. - From $b$ to $c$, there is 1 edge ($e_5$). - The loop $e_1$ on $a$ cannot be used because it would repeat vertex $a$. **Number of paths:** $3 \times 1 = 3$ --- 4. **Counting Trails from $a$ to $c$:** - Trails cannot repeat edges but can repeat vertices. - The loop $e_1$ on $a$ can be used once or not at all. - From $a$ to $b$, we have 3 edges ($e_2$, $e_3$, $e_4$), each can be used once. - From $b$ to $c$, edge $e_5$ can be used once. Possible trails: - Use $e_1$ zero or one time (loop on $a$). - Then choose one of the 3 edges from $a$ to $b$. - Then use $e_5$ to $c$. Number of trails: - Without loop: $3$ trails - With loop once: $3$ trails (loop $e_1$ then one of $e_2$, $e_3$, $e_4$, then $e_5$) Total trails: $3 + 3 = 6$ --- 5. **Counting Walks from $a$ to $c$:** - Walks can repeat edges and vertices any number of times. - The loop $e_1$ on $a$ can be used any number of times. - The edges $e_2$, $e_3$, $e_4$ from $a$ to $b$ can be used any number of times. - Edge $e_5$ from $b$ to $c$ can be used once to reach $c$. Since the graph is finite and the loop on $a$ can be repeated infinitely, there are infinitely many walks from $a$ to $c$. --- **Final answers:** - Paths from $a$ to $c$: $3$ - Trails from $a$ to $c$: $6$ - Walks from $a$ to $c$: infinitely many