The problem is asking for the number of different ways to get from point A to point R using the shortest path possible by moving along the grid lines in the figure.

Key Observations:

1. The path from point A to point R consists of a combination of right and down movements.
2. To determine the shortest path, count the number of right (R) and down (D) steps needed:
   • From the image, you need to move 3 steps to the right and 3 steps down.

Mathematical Approach:

This is a typical combinatorics problem where we need to determine how to arrange a certain number of movements.

• You need to make a total of 6 moves (3 right and 3 down).
• The number of different ways to arrange these moves is the number of ways to choose 3 positions (for either the right or the down movements) out of the 6 total moves. This is given by the binomial coefficient:

$\binom{6}{3} = \frac{6!}{3!3!}$

Let me calculate this for you.The number of different ways to get from point A to point R using the shortest path possible is 20.

Would you like a more detailed explanation, or do you have any further questions?

Here are 5 related questions you might find helpful:

1. How do binomial coefficients apply in other path-counting problems?
2. What are some real-world applications of combinatorics in grid-based problems?
3. Can you solve a similar problem if the grid had more rows or columns?
4. What if you could move diagonally? How would that affect the number of paths?
5. How does Pascal's triangle relate to counting paths in a grid?

Tip: When dealing with path-counting problems, always check for symmetry and determine if there are shortcuts using combinations.