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The image you've uploaded appears to deal with different counting strategies in mathematics, particularly using combinatorics and basic arithmetic operations to solve problems. Here’s a breakdown of the two main problems presented:

### Problem 1: Path Counting on a Grid
The problem asks for the number of distinct paths between points A, B, and C on a grid:
- **A to B:** There are 4 distinct paths shown.
- **B to C:** There are 6 distinct paths.
- **A to C via B:** The problem combines these paths using the multiplication principle of counting: There are \(4 \times 6 = 24\) ways to travel from A to C passing through B.

This uses a method from combinatorics, which involves counting minimal distance paths on a grid.

### Problem 2: Chess Tournament Pairing
In this problem, 6 friends organize a chess tournament. The question asks for the total number of games played in a double round-robin format.
- A double round-robin means each pair of players plays two games (one as white and one as black).
- The number of distinct pairs in a group of 6 is calculated using the combination formula \( \binom{6}{2} = \frac{6 \times 5}{2} = 15 \) pairs.
- Since each pair plays two games, the total number of games is \( 15 \times 2 = 30 \).

### General Tip:
When calculating paths on a grid or pairings in a group, combinatorics offers useful tools such as the multiplication principle (for paths) and combination formulas (for pairings).

Do you want more details on any part of the solutions, or do you have any questions?

Here are 5 related questions you can explore:
1. How can you calculate the number of paths in larger grids?
2. What is the difference between permutations and combinations?
3. How would you approach this problem if diagonals were allowed as valid paths?
4. How does the formula for combinations work in general situations?
5. How would the solution change if the chess tournament involved only single games?

**Tip**: Whenever counting paths or pairings, always verify if the problem involves repetition or special constraints, as this impacts whether you use combinations or permutations.

¿Cuántos caminos de distancia mínima van de A a B, de B a C, y cuántos van de A a C pasando por B? Además, ¿cuántas partidas de ajedrez han de jugar en total en un torneo de 6 amigos a doble vuelta?

Learn how to count paths on a grid using combinatorics and calculate total games in a chess tournament. Explore methods for solving minimal distance path problems and pairings in a double round-robin format. Suitable for high school students.

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