For options A, B, C, and D, how to draw visual diagrams for all valid path scenarios that meet the conditions? For each distinct case, draw a detailed diagram, and use these visual illustrations to help understand and solve the problem.

(*Find all valid non-negative integer solutions*)sol = 
 FindInstance[{x + y + z + w == 4, x - y == -2, w == z, x >= 0, 
   y >= 0, z >= 0, w >= 0}, {x, y, z, w}, Integers, 10]

(*Calculate number of paths for each solution*)
paths = 4!/(x! y! z! w!) /. sol

(*Total number of valid paths*)
total = Total[paths]

(*Check if option A is correct*)
Print["Is option A correct? ", total == 12]

The above is the computational method I used to verify whether Option A is correct. Now I want to visualize all valid paths for each option using grid diagrams, count the number of paths and calculate the probabilities to determine the correctness of each option. How can I draw such diagrams?