Example 4 Fill 2 $a$s and 2 $b$s, a total of 4 letters, in the 16 small squares of a $4 \times 4$ grid, with at most 1 letter in each small square. If the same letters are neither in the same row nor in the same column, then the number of different ways to fill the grid is (answer with a number).
(2007 National High School League Question)
To make 2 $a$s neither in the same row nor in the same column, there are $C_{4}^{2} P_{4}^{2}=72$ ways. Similarly, to make 2 $b$s neither in the same row nor in the same column, there are also $C_{4}^{2} P_{4}^{2}=72$ ways. Therefore, by the multiplication principle, there are $72^{2}$ such ways. Among them, there are two cases that do not meet the requirements: the case where the 2 $a$s are in the same cells as 2 $b$s has 72 ways; the case where only 1 of the 2 $a$s is in the same cell as a $b$ has $C_{16}^{1} P_{9}^{2}=16 \times 72$ ways. Therefore, the number of ways that meet the conditions of the problem is $72^{2}-72-16 \times 72=55 \times 72=3960$ ways.