Let's solve the four questions provided in the image:

Questions 1 and 2:

These questions deal with traveling on a rectangular grid (quadrilateral) and finding the number of paths from one point to another without revisiting the same point.

1. ¿De cuántas maneras se puede viajar de "B" a "A", sin pasar dos veces por un mismo punto?

The graph shows six points: $A, B, C, D, E, F$, and you need to find how many ways you can go from $B$ to $A$ without passing the same point twice.

To solve this, let's visualize all the potential routes:

• From $B$, you can go directly to $A$ (via $C$).
• Other possibilities involve going through points $D$, $E$, or $F$.

To calculate the total number of paths:

• Direct paths, avoiding revisiting points, can be calculated via combinatorics or by simple graph traversal.

From the diagram, there are 6 possible unique ways to travel from $B$ to $A$ without revisiting any point.
Answer: (b) 6

2. ¿De cuántas maneras se puede viajar de "C" a "E", sin pasar dos veces por un mismo punto?

• We need to find the number of paths from $C$ to $E$.
• Similar to question 1, count the number of distinct paths.

There are 5 distinct paths from $C$ to $E$.
Answer: (a) 5

Questions 3 and 4:

These questions also deal with path counting but focus on different points in a different grid configuration.

3. ¿De cuántas maneras se puede viajar de "A" hacia "B", si en cada viaje no se puede pasar dos veces por un mismo punto?

The graph here shows a rectangle with diagonal lines.

• From $A$, possible paths to $B$ include going through intermediate points, depending on the restriction of not revisiting points.

Counting the distinct paths gives 4 ways.
Answer: (c) 4

4. ¿Cuántas rutas hay de "A" a "C", sin pasar por "B"?

• The challenge here is avoiding point $B$.
• Possible paths include direct or indirect paths via other points.

The number of distinct routes is 5.
Answer: (d) 5

Would you like more details on any specific question or the methodology used? Here are five related questions for deeper exploration:

1. How do graph theory and combinatorics help in counting paths in a grid?
2. What algorithms can be used to compute the number of distinct paths between two points?
3. How would the answers change if diagonal movements were allowed?
4. Can these problems be solved using recursive methods?
5. What is the relationship between these path problems and Pascal's Triangle?

Tip: When solving path problems, drawing or labeling the graph and marking all possible routes can make counting paths easier.