The problem asks for the number of different ways to reach point $R$ from point $A$ by following the lines in the given grid, taking the shortest path.

Analyzing the problem:

The grid shows a path from $A$ to $R$, and we can observe that to go from $A$ to $R$, a minimum number of moves must be made. These moves consist of:

• $4$ moves to the right (denoted as $R$),
• $4$ moves downwards (denoted as $D$).

Thus, any shortest path from $A$ to $R$ is composed of exactly $4$ rightward moves and $4$ downward moves. The problem then reduces to counting the number of distinct ways to arrange these moves.

Solution approach:

This problem can be solved using combinatorics. Specifically, the number of distinct paths is the number of ways to arrange $4$ $R$'s (right moves) and $4$ $D$'s (downward moves) in a sequence of $8$ moves. This is a typical application of the binomial coefficient:

$\text{Number of paths} = \binom{8}{4} = \frac{8!}{4!4!}$

Now, I'll calculate the value.

Calculation:

$\binom{8}{4} = \frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1} = 70$

So, the correct number of shortest paths is $70$, but this isn't one of the options given. Let me reconsider the path structure, as there might be additional restrictions or specific grid behavior in the problem that affects the final count.

Would you like a detailed explanation of a potential alternative interpretation?