The following figure shows a $3^2 \times 3^2$ grid divided into $3^2$ subgrids of size $3 \times 3$. This grid has $81$ cells, $9$ in each subgrid.
[asy]
draw((0,0)--(9,0)--(9,9)--(0,9)--cycle, linewidth(2));
draw((0,1)--(9,1));
draw((0,2)--(9,2));
draw((0,3)--(9,3), linewidth(2));
draw((0,4)--(9,4));
draw((0,5)--(9,5));
draw((0,6)--(9,6), linewidth(2));
draw((0,7)--(9,7));
draw((0,8)--(9,8));
draw((1,0)--(1,9));
draw((2,0)--(2,9));
draw((3,0)--(3,9), linewidth(2));
draw((4,0)--(4,9));
draw((5,0)--(5,9));
draw((6,0)--(6,9), linewidth(2));
draw((7,0)--(7,9));
draw((8,0)--(8,9));
[/asy]
Now consider an $n^2 \times n^2$ grid divided into $n^2$ subgrids of size $n \times n$. Find the number of ways in which you can select $n^2$ cells from this grid such that there is exactly one cell coming from each subgrid, one from each row and one from each column.
To solve this problem, we need to find the number of ways to select \( n^2 \) cells from an \( n^2 \times n^2 \) grid such that there is exactly one cell from each subgrid, one from each row, and one from each column. 

1. **Understanding the Grid Structure**:
   - The grid is divided into \( n^2 \) subgrids, each of size \( n \times n \).
   - Each subgrid contains \( n \) rows and \( n \) columns.
   - We need to select one cell from each subgrid such that no two selected cells are in the same row or column of the entire \( n^2 \times n^2 \) grid.

2. **Counting the Selections**:
   - We can think of this problem as a permutation problem where we need to place \( n \) cells in \( n \) rows and \( n \) columns such that no two cells are in the same row or column.
   - This is equivalent to finding a permutation of \( n \) elements for each subgrid.

3. **Permutations within Subgrids**:
   - For each subgrid, there are \( n! \) ways to arrange \( n \) cells such that no two cells are in the same row or column within the subgrid.
   - Since there are \( n^2 \) subgrids, and each subgrid's arrangement is independent of the others, we have \( (n!)^{n^2} \) ways to arrange the cells within the subgrids.

4. **Ensuring Global Constraints**:
   - The arrangement within each subgrid must also satisfy the global constraint that no two cells are in the same row or column of the entire \( n^2 \times n^2 \) grid.
   - This is equivalent to finding a permutation of \( n^2 \) elements where each element represents a subgrid, and the permutation ensures that no two elements (subgrids) are in the same row or column of the entire grid.

5. **Combining the Counts**:
   - The number of ways to arrange \( n^2 \) subgrids such that no two subgrids are in the same row or column is \( (n^2)! \).
   - Therefore, the total number of ways to select \( n^2 \) cells from the grid is the product of the number of ways to arrange the cells within the subgrids and the number of ways to arrange the subgrids themselves.

6. **Final Calculation**:
   \[
   \text{Total number of ways} = (n!)^{n^2} \times (n^2)!
   \]

Thus, the number of ways to select \( n^2 \) cells from the \( n^2 \times n^2 \) grid such that there is exactly one cell from each subgrid, one from each row, and one from each column is given by:

\[
\boxed{(n!)^{n^2} \times (n^2)!}
\]