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 $6+6+9+9+9+9+6+6 = 60$.
