Introduction:
To determine the number of determinants with a value of zero for all A and B in the elements of a third-order determinant, we need to consider the properties of determinants and apply them to the given scenario.

Properties of Determinants:
1. If two rows or two columns of a determinant are identical, then the determinant value is zero.
2. Swapping two rows or two columns of a determinant changes the sign of the determinant.
3. If a determinant has a row or column of zeros, then the determinant value is zero.

Analysis:
We have a third-order determinant, which means it has 3 rows and 3 columns. Let's denote the determinant as D.

D = |a1 a2 a3|
|b1 b2 b3|
|c1 c2 c3|

To find the number of determinants with a value of zero for all A and B, we need to consider the possibilities that can lead to a determinant value of zero.

Case 1: Two Rows or Two Columns are Identical:
If two rows or two columns are identical, then the determinant value is zero. In this case, we have 3 choices for the first row, 3 choices for the second row, and 1 choice for the third row (since it needs to be identical to one of the previous rows). Similarly, we have 3 choices for the first column, 3 choices for the second column, and 1 choice for the third column. Therefore, the total number of determinants in this case is 3 * 3 * 1 * 3 * 3 * 1 = 81.

Case 2: A Row or Column is Zero:
If a determinant has a row or column of zeros, then the determinant value is zero. In this case, we have 3 choices for the row/column that will be zero, and 2 choices for the remaining rows/columns. Therefore, the total number of determinants in this case is 3 * 2 = 6.

Conclusion:
The total number of determinants with a value of zero for all A and B is the sum of the determinants from Case 1 and Case 2, which is 81 + 6 = 87.

Answer:
None of the given options (a), (b), (c) match the calculated value of 87. Therefore, the correct answer is (d) none of these.

Correct is C