help asap!!

Let's focus on the letter O's.   First note that the middle O cannot participate in any valid ROOM path, so we may as well get rid of it.   Now create a graph by letting each letter O be a vertex, and joining two neighboring vertices if we can go from one O to another by taking one step in any direction (up, down, left, right, or diagonal).

Here is the picture of the graph where each o is a vertex.  The edges are represented by the sequence of dashes or vertical line.   We also name the edges by the letters from A to H.

          A      B
      o-----o-----o
   C |                  | D
      o                 o
   E |                  | F
      o-----o-----o
          G     H
 

Now, to each vertex let's write down the number of the letters R and M it can go to in one step.   For example, entry 3r,2m means the vertex at that position can go to 3 different R's in one step, and 2 different M's in one step, etc.

    3r,2m 3r,0m 3r,2m
    0r,3m           0r,3m
    3r,2m 3r,0m 3r,2m
 

Now, to each edge we count how many ways it can be part of a ROOM path.  It is part of a ROOM path iff one end vertex is connected to R and the other end vertex is connected to M.   We show the calculation below.

    A: 3 * 2 = 6
    B: 3 * 2 = 6
    C: 3 * 3 = 9
    D: 3 * 3 = 9
    E: 3 * 3 = 9
    F: 3 * 3 = 9
    G: 3 * 2 = 6
    H: 3 * 2 = 6
 

So the total number of ways equals $6+6+9+9+9+9+6+6 = 60$.
 

Here is a similar problem   : https://web2.0calc.com/questions/help_63419#r1

Very impressive, Bginner   !!!!

No clue on my part   .....LOL!!!!!