+73 How many continuous paths from $A$ to $B$, along segments of the figure, do not revisit any of the six labeled points?
+876 Since there are only two ways to end to $B$, we have 2 cases.
Case 1: It ends with $\overline{FB}.$
There are 6 paths: $A \Rightarrow C \Rightarrow F \Rightarrow B$
$A \Rightarrow C \Rightarrow D \Rightarrow E \Rightarrow F \Rightarrow B$
$A \Rightarrow D \Rightarrow C \Rightarrow F \Rightarrow B$
$A \Rightarrow D \Rightarrow E \Rightarrow F \Rightarrow B$
$A \Rightarrow D \Rightarrow F \Rightarrow B$
$A \Rightarrow C \Rightarrow D \Rightarrow F \Rightarrow B.$
Case 2: It ends with $\overline{CB}.$
There are 4 paths: $A \Rightarrow D \Rightarrow C \Rightarrow B$
$A \Rightarrow D \Rightarrow F \Rightarrow C \Rightarrow B$
$A \Rightarrow C \Rightarrow B$
$A \Rightarrow D \Rightarrow E \Rightarrow F \Rightarrow B$
Adding these cases yields $6+4=\boxed{10 \text{ ways.}}$
If anyone finds a better way, please hit me up as this casework a bit tedious to do.
- Jimmy
