In the $4 \times 4$ array shown below, each cell of the first three rows has either a cross (X) or a number. The number in a cell represents the count of the immediate neighboring cells (left, right, top, bottom, diagonals) NOT having a cross (X). Given that the last row has no crosses (X), the sum of the four numbers to be filled in the last row is

Let the 4 cells in the fourth (last) row from left to right be denoted as $c_1, c_2, c_3, c_4$.

We are given that no cell in the fourth row contains a cross (X). Therefore, all neighbors within row 4 do not have crosses.

Let us calculate the count of immediate non-cross neighbors for each cell in row 4:

1. Cell $c_1$ (row 4, column 1):

   • Neighboring cells: $(3,1)$, $(3,2)$, and $(4,2)$. (Total = 3 neighbors)

   • Non-cross neighbors: $(3,1)$ which is $3$, and $(4,2)$ which is a number.

   • Cross neighbor: $(3,2)$ which is $\text{X}$.

   • Value of $c_1 = 2$.

2. Cell $c_2$ (row 4, column 2):

   • Neighboring cells: $(3,1)$, $(3,2)$, $(3,3)$, $(4,1)$, and $(4,3)$. (Total = 5 neighbors)

   • Non-cross neighbors: $(3,1)$, $(3,3)$, $(4,1)$, and $(4,3)$.

   • Cross neighbor: $(3,2)$ which is $\text{X}$.

   • Value of $c_2 = 5 - 1 = 4$.

3. Cell $c_3$ (row 4, column 3):

   • Neighboring cells: $(3,2)$, $(3,3)$, $(3,4)$, $(4,2)$, and $(4,4)$. (Total = 5 neighbors)

   • Non-cross neighbors: $(3,3)$, $(4,2)$, and $(4,4)$.

   • Cross neighbors: $(3,2)$ which is $\text{X}$, and $(3,4)$ which is $\text{X}$.

   • Value of $c_3 = 5 - 2 = 3$.

4. Cell $c_4$ (row 4, column 4):

   • Neighboring cells: $(3,3)$, $(3,4)$, and $(4,3)$. (Total = 3 neighbors)

   • Non-cross neighbors: $(3,3)$ and $(4,3)$.

   • Cross neighbor: $(3,4)$ which is $\text{X}$.

   • Value of $c_4 = 3 - 1 = 2$.

Sum of the four numbers filled in the last row:

Thus, option (A) is correct.