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To solve this problem, we need to find the number of shortest paths from point \( A \) to point \( R \) in the grid. Here is how we can approach it:

### Observing the Grid:
- We can only move **down** or **right** along the lines in the grid.
- The shortest path consists of the minimum number of moves, where each move takes us closer to \( R \) either horizontally or vertically.
- We need to calculate how many down and right moves are required to get from \( A \) to \( R \).

### Step 1: Number of Moves
- From \( A \), we need to move **down** and **right** to get to \( R \).
- Count the vertical lines from \( A \) to \( R \):
  - There are **3 vertical moves** (3 "down" steps).
  
- Count the horizontal lines from \( A \) to \( R \):
  - There are **4 horizontal moves** (4 "right" steps).
  
Thus, the shortest path requires exactly **3 down moves** and **4 right moves**.

### Step 2: Total Number of Paths
The number of different paths can be found by determining how many ways we can arrange these moves. Since we are making a total of \( 7 \) moves (3 downs and 4 rights), the problem boils down to choosing 3 positions for the "down" moves out of 7.

This is a combinatorial problem, and the number of such arrangements is given by the combination formula:
\[
\text{Number of paths} = \binom{7}{3} = \frac{7!}{3!(7-3)!} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35
\]

### Final Answer:
There are **35 different shortest paths** from point \( A \) to point \( R \).

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Would you like more details on the combinatorics involved or have any other questions?

Here are 5 questions related to this concept:
1. What is the general formula for the number of shortest paths on a grid?
2. How does the number of paths change if we increase the grid size?
3. Can this method be applied to three-dimensional grids? If so, how?
4. What if diagonal moves were allowed in the grid—how would that affect the calculation?
5. How does this concept connect to Pascal's triangle?

**Tip:** In combinatorics problems, identifying the number of each type of required move (right/down) and their arrangement is key to calculating the number of possible paths.

In how many different ways can you get from point A to point R using the shortest path possible, by moving along the lines in the figure?

Learn how to calculate the number of shortest paths from point A to point R on a grid by using combinatorics. This problem involves counting the number of right and down moves and applying the combination formula. Suitable for students in Grades 7-10.

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