1.113 Let a large cube of $4 \times 4 \times 4$ be composed of 64 unit cubes. Select 16 of these unit cubes to be painted red, such that in each of the $1 \times 1 \times 4$ small columns of the large cube, there is exactly one red unit cube. How many different ways are there to select the 16 red cubes? Explain your reasoning.
[Solution 1] Divide the large cube into 4 layers in the order from front to back, each layer being a 1 × 4 × 4 rectangular prism. Number the 4 layers as 1, 2, 3, 4. Project all the red cubes onto the 4 × 4 square grid on the front face. Thus, the projections of the 16 red cubes are all different and exactly fill the 4 × 4 square. Write 1, 2, 3, or 4 in each unit square to indicate which layer the red cube is from. In this way, the different ways to choose the red cubes correspond one-to-one with the different ways to fill the 4 × 4 square grid with numbers. Therefore, we only need to calculate the number of different ways to fill the grid.

It is easy to see that each of the 16 unit cubes in each layer contains exactly 4 red cubes, and these 4 red cubes are in different rows and columns. Therefore, in the 4 × 4 square, the 16 numbers filled in include 4 each of 1, 2, 3, and 4, and each set of 4 identical numbers is in different rows and columns.

First, fill the 4 ones into the 4 × 4 grid, each in a different row and column, which can be done in 4! = 24 different ways.

Then, note that the other 3 numbers in the first row can only be 2, 3, and 4, each appearing once, which can be done in 6 different ways.

For any way of filling the 4 ones, it is always possible to swap layers so that the 4 ones are exactly on the main diagonal of the 4 × 4 grid. Then, by permuting the last 3 numbers in the first row, we can assume the first row is 1, 2, 3, 4.

Consider the square in the second row and first column, which can be filled with 2, 3, or 4. If it is filled with 2, then the 4 numbers in the second row are 2, 1, 4, 3. Thus, the squares in the third row and fourth column and the fourth row and third column must be 2. The remaining 2 × 2 square in the lower left must contain 3 and 4, each appearing twice and with the same number on the diagonal, which can be done in 2 different ways. If the square in the second row and first column is filled with 3, then the 4 numbers in the second row are 3, 1, 4, 2, and there is only one way to fill the rest. Similarly, if the square in the second row and first column is filled with 4, there is also only one way to fill the rest. Therefore, there are 4 different ways to fill the grid.

In summary, there are 24 × 6 × 4 = 576 different ways to fill the grid. Thus, there are 576 different ways to choose the 16 red cubes.
[Solution 2] From Solution 1, we know that each valid way to choose the 16 red cubes corresponds to a way to divide the 4 × 4 grid into 4 groups, each group containing 4 squares in different rows and columns. Therefore, we only need to count the number of different ways to divide the grid into groups.

Use 1, 2, 3, 4 to represent the group number of each square and fix the 4 squares in the first column to belong to groups 1, 2, 3, 4, respectively. Thus, there are 24 different ways to divide the grid:

By corresponding the 4 groups of squares to the 4 layers of red cubes, there are 24 different ways to do so. Therefore, there are 576 different ways to choose the 16 red cubes.