Introduction:
The given equation is abc = 30, where a, b, and c are positive integers. We need to find the number of positive integral solutions for this equation.

Approach:
To find the number of positive integral solutions, we can factorize the number 30 into its prime factors.

Factorizing 30:
30 can be factorized as follows:

30 = 2 × 3 × 5

Now, we can consider the factors of 30, i.e., 2, 3, and 5, as the possible values for a, b, and c. Let's analyze the possibilities.

Possible values for a, b, and c:
We have the prime factors of 30 as 2, 3, and 5. Now, let's consider the possible values for a, b, and c.

Since a, b, and c are positive integers, we can consider the following cases:

1. Case 1: a = 2, b = 3, and c = 5
In this case, we have a solution for abc = 30.

2. Case 2: a = 2, b = 5, and c = 3
In this case, we have another solution for abc = 30.

3. Case 3: a = 3, b = 2, and c = 5
In this case, we have another solution for abc = 30.

4. Case 4: a = 3, b = 5, and c = 2
In this case, we have another solution for abc = 30.

5. Case 5: a = 5, b = 2, and c = 3
In this case, we have another solution for abc = 30.

6. Case 6: a = 5, b = 3, and c = 2
In this case, we have another solution for abc = 30.

Total number of solutions:
From the above analysis, we can see that there are 6 different solutions for abc = 30. Therefore, the number of positive integral solutions for this equation is 6.

Conclusion:
The correct answer is option 'C' - 27.