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The problem you have uploaded involves calculating the number of shortest routes between two points, A and B, where participants must descend one building, move across, and ascend another building. This type of problem involves counting paths on a grid using combinatorics, particularly using binomial coefficients, since it restricts movement to horizontal and vertical directions.

### Problem Overview:
1. Participants must move from point A to point B.
2. They are required to move down the first building, cross the space between the buildings, and then climb up the second building.
3. Movements are restricted to horizontal and vertical directions, so the task is to find how many shortest routes are possible.

### Steps to Solve:
This problem can be thought of as a grid-walking problem where you need to find the number of ways to reach point B from point A, moving down, across, and then up. For a grid problem like this, we can use combinations to count the distinct paths.

1. **Define the grid structure**:
   - Let’s count the number of horizontal and vertical steps required between points A and B.
   - Vertical moves are either up or down, while horizontal moves are strictly between the two buildings.

2. **Calculate total steps**:
   - First building (from A downwards): count the vertical steps.
   - Horizontal steps between buildings.
   - Second building (upwards to B): count the vertical steps.

Once I analyze the exact grid size based on the image and problem, I can compute the total number of possible shortest routes using combinations.

Let me analyze the structure of the grid more carefully, and I'll proceed with the calculation!I've opened the image to better analyze the grid. Let's now break down the steps based on the vertical and horizontal movements in the image:

- The first building has 3 vertical steps down.
- The horizontal space between the two buildings has 4 horizontal steps.
- The second building has 2 vertical steps up.

### Now, to find the total number of shortest routes:
1. You need 3 downward vertical moves on the first building.
2. You need 4 horizontal moves to cross between the buildings.
3. You need 2 upward vertical moves to reach point B.

This results in a total of:
- 3 vertical downward moves,
- 4 horizontal moves, and
- 2 vertical upward moves.

This makes a total of \( 3 + 4 + 2 = 9 \) steps, composed of 3 down, 4 right, and 2 up.

The total number of distinct paths is given by the combination formula \( C(n, k) \), where \( n \) is the total number of steps, and \( k \) is the number of down, up, or horizontal steps.

I'll now calculate this.The total number of shortest routes from point A to point B is **1,260**. This includes all combinations of downward, horizontal, and upward movements between the two buildings.

Would you like a more detailed breakdown of the solution or any other related questions?

Here are 5 additional questions you might find interesting:
1. How would the number of paths change if the second building had more vertical steps to climb?
2. What if the horizontal distance between the buildings was reduced by one step?
3. How can this problem be extended to three buildings?
4. How would the result differ if diagonal movement was allowed?
5. What are the computational limits for much larger grids?

**Tip:** For grid-based problems, the combination formula \( C(n, k) \) helps in finding the number of ways to choose steps, especially when movement is restricted to specific directions.

There are 2 buildings that are currently being constructed side by side. A participant needs to climb down from point A, go to the other building, and climb up to point B. The participant is required to walk along any horizontal path and climb down/up along any vertical path. How many shortest routes are there from A to B?

Explore the shortest routes between two buildings in a grid layout using combinatorial mathematics. This problem involves path counting techniques and the use of the binomial coefficient to calculate the number of possible shortest paths. Suitable for students in Grades 9-12.

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