Introduction:
We are given the equation 7x + 12y + 4z = 30, where x, y, and z are positive integers. We need to determine the number of positive integral solutions that satisfy this equation.

Approach:
To find the number of positive integral solutions, we can use the concept of Diophantine equations. We will solve the given equation step by step to arrive at the solution.

Solution:

Step 1: Finding a particular solution:
Let's start by finding a particular solution to the given equation. We can assume x = 0 and solve for y and z.

7x + 12y + 4z = 30
Substituting x = 0,
12y + 4z = 30
Dividing by 2,
6y + 2z = 15

This equation has a solution when y = 2 and z = 3.
So, a particular solution is x = 0, y = 2, and z = 3.

Step 2: Finding the general solution:
To find all the solutions, we need to find the general solution by adding a solution to the homogeneous equation (where the right-hand side is 0) to the particular solution found in step 1.

The homogeneous equation is:
7x + 12y + 4z = 0

To find a solution to this equation, we can assume x = 4, y = -7, and z = 3. These values satisfy the homogeneous equation.

Now, we can add this solution to the particular solution:
x = 0 + 4 = 4
y = 2 + (-7) = -5
z = 3 + 3 = 6

So, the general solution is x = 4, y = -5, and z = 6.

Step 3: Checking for positive integral solutions:
We need to check if the general solution obtained in step 2 satisfies the condition of positive integers.

x = 4, y = -5, and z = 6 are not positive integers, so they are not valid solutions.

Therefore, there is only one positive integral solution to the given equation.

Conclusion:
The number of positive integral solutions for the equation 7x + 12y + 4z = 30 is 1.