The question is asking for the number of different shortest paths from point A to point R in a grid-like figure.

Solution:

• The movement in the grid is restricted to rightward or downward steps, as these are the only directions that move closer to point R.
• To reach R from A, you need to count how many rightward and downward steps are required.

Counting steps:

From the diagram:

• There are 5 rightward steps (horizontal moves).
• There are 4 downward steps (vertical moves).

The total number of steps required is $5 + 4 = 9$ steps. Out of these, 5 steps will be rightward (R), and 4 steps will be downward (D).

The number of different paths corresponds to the number of ways to arrange 5 R's and 4 D's in a sequence of 9 steps. This is a combinatorics problem, where we calculate the number of ways to choose 5 positions for the rightward moves (or equivalently 4 positions for the downward moves) from 9 available positions.

The formula for this is: $\text{Number of paths} = \binom{9}{5} = \binom{9}{4} = \frac{9!}{5!(9-5)!} = \frac{9!}{5!4!}$

Let's calculate that.The number of different shortest paths from point A to point R is 126.

It seems that none of the answer choices (A: 42, B: 47, C: 52, D: 57, E: 62) match this result, which suggests there may be an issue with either the question or the provided options.

Would you like further clarification or help with anything else?

Here are 5 related questions for deeper understanding:

1. How does the combination formula apply to calculating paths in a grid?
2. What would happen if diagonal moves were allowed in the problem?
3. How can Pascal's Triangle help solve grid path problems?
4. How would the solution change if obstacles were placed in the grid?
5. What is the general formula for counting paths in an $m \times n$ grid?

Tip: When solving combinatorics problems involving paths, visualize the steps and use the combination formula to select specific directions from the total moves.