Activity $4\mathrm{.}1\mathrm{.}1$: Counting with Addition and Multiplication

The graph below $($called a "word$-$chain automaton"$)$ defines a collection of sentences. To form a sentence using this graph, start at vertex v$0$ and follow a directed path to vertex $.$ The sentence is then formed by the words on the edges of this path. For example, the sentence "Moe ate green bananas" results from following the edges along the top of the graph.

How many different sentences can be formed in this way? Express your answer as an numeric expression that shows how addition and multiplication are used in this counting problem.

Create a simple word$-$chain automaton that produces exactly $19$ sentences.