5. (10 points) The grid in the figure is composed of 6 identical small squares. Color 4 of the small squares gray, requiring that each row and each column has a colored small square. Two colored grids are considered the same if they are identical after rotation. How many different coloring methods are there? $\qquad$ kinds.
【Analysis】First, we can classify the discussion based on the number of colored cells in the first column, noting that rotations resulting in the same pattern are considered the same coloring method.

【Solution】Solution: (1) When the first column has 3 colored cells, the coloring situations are as follows: there are 3 cases;
(2) When the first column has 2 colored cells, the coloring situations are as follows: there are 4 cases.
In total, $3+4=7$ cases.
Therefore, the answer is: 7.