To find the number of matrices of rank three in the space M, we need to determine the number of ways we can choose the entries of the matrix to ensure it has rank three.

Let's break down the process step by step:

Step 1: Choosing the first three rows
Since the rank of the matrix is three, the first three rows must be linearly independent. The first row has 4 choices (any non-zero vector in the finite field of three elements). For the second row, we cannot choose a vector that is a linear combination of the first row, so we have 3 choices. Similarly, for the third row, we cannot choose a vector that is a linear combination of the first two rows, so we have 3 choices again.

Number of ways to choose the first three rows = 4 * 3 * 3 = 36

Step 2: Choosing the fourth row
The fourth row must be a linear combination of the first three rows. Since the rank of the matrix is three, there must be at least one row that is not a linear combination of the other two. Let's consider the three possible cases:

Case 1: The fourth row is a linear combination of the first row, but not the second or third row.
In this case, the fourth row cannot be the zero vector, so we have 3 choices for the fourth row.

Case 2: The fourth row is a linear combination of the second row, but not the first or third row.
Similar to case 1, we have 3 choices for the fourth row.

Case 3: The fourth row is a linear combination of the third row, but not the first or second row.
Again, we have 3 choices for the fourth row.

Number of ways to choose the fourth row = 3 + 3 + 3 = 9

Step 3: Multiplying the choices
To find the total number of matrices of rank three, we multiply the number of choices in each step.

Total number of matrices of rank three = (number of ways to choose the first three rows) * (number of ways to choose the fourth row)
= 36 * 9 = 324

Step 4: Simplifying the expression
The given options are in terms of (34 - k), where k is a positive integer. We can simplify the expression to match one of the options:

324 = (34 - 1) * (34 - 3) * (34 - 32)

Therefore, the correct answer is option C) (34 - 1)(34 - 3)(34 - 32).