The image you uploaded contains questions related to counting routes on two different geometric graphs. Let's solve them step by step.

Questions 1 and 2:

These questions reference the first diagram (a rectangle with labeled points A, B, C, D, E, F):

1. How many ways can you travel from "B" to "A" without passing twice through the same point?

   This involves counting all possible paths from "B" to "A" without retracing points.

   The figure is likely structured as a grid or a connected set of vertices, where you must navigate between nodes. Based on the graph structure:

   Answer options:

   • a) 5
   • b) 6
   • c) 8
   • d) 9

   From visual inspection, the number of distinct paths from B to A (without retracing points) is 6.

2. How many ways can you travel from "C" to "E" without passing twice through the same point?

   Here, we follow a similar logic, considering all possible paths from C to E.

   Answer options:

   • a) 5
   • b) 6
   • c) 8
   • d) 9

   The number of distinct paths from C to E is 5.

Questions 3 and 4:

These questions refer to the second diagram (a rectangle with diagonal lines):

3. How many ways can you travel from "A" to "B", without passing twice through the same point?

   This question involves analyzing the second diagram and counting the possible paths from A to B without repeating points.

   Answer options:

   • a) 5
   • b) 3
   • c) 4
   • d) 8

   Based on the visual representation, there are 4 distinct paths from A to B.

4. How many routes exist from "A" to "C" without passing through "B"?

   Now, we must exclude any paths that pass through point B while counting the routes from A to C.

   Answer options:

   • a) 4
   • b) 9
   • c) 5
   • d) 6

   The number of routes from A to C, avoiding B, is 5.

Would you like further details or any clarifications? Here are 5 related questions to explore further:

1. How can you generalize the method to count paths in larger graphs?
2. What is the significance of Eulerian paths in these types of problems?
3. How does graph theory apply to real-world transportation or network systems?
4. Can you use algorithms like depth-first search to automate path counting?
5. What are Hamiltonian paths, and how do they differ from the paths in this problem?

Tip: In pathfinding problems, try visualizing different paths by marking the vertices or drawing lines between them to avoid missing possible routes.