Suppose I would like to randomly fill an empty matrix (consisting of zeroes) with a particular number of each of the elements $a$ and $b$. One way is first fill the matrix with the proper number of $a$'s and then with the proper number of $b$'s. Another way is to fill the $b$'s first and then the $a$'s. My question is whether the order affects the distribution of the resulting matrices. How about for a collection of $n$ elements (assuming the matrix is large enough to do so)?
For example, below is a filling of a $3\times 3$ matrix with $3$ $a$'s and $2$ $b$'s. At each step the entry is determined by a uniform distribution on the set of empty entries. We may fill the $a$'s first: $$\left[\begin{matrix}\_&\_&\_\\\ \_&\_&\_\\\ \_&\_&\_\\\end{matrix}\right]\overset{\text{Unif}(9)}\mapsto\left[\begin{matrix}\_&\_&\_\\\ a&\_&\_\\\ \_&\_&\_\\\end{matrix}\right]\overset{\text{Unif}(8)}\mapsto\left[\begin{matrix}\_&a&\_\\\ a&\_&\_\\\ \_&\_&\_\\\end{matrix}\right]\overset{\text{Unif}(7)}\mapsto\left[\begin{matrix}\_&a&\_\\a&a&\_\\ \_&\_&\_\end{matrix}\right]$$ $$\overset{\text{Unif}(6)}\mapsto\left[\begin{matrix}\_&a&\_\\a&a&\_\\ \_&b&\_\end{matrix}\right]\overset{\text{Unif}(5)}\mapsto\left[\begin{matrix}\_&a&\_\\a&a&b\\ \_&b&\_\end{matrix}\right]$$
Alternatively we may fill the $b$'s first:
$$\left[\begin{matrix}\_&\_&\_\\\ \_&\_&\_\\\ \_&\_&\_\\\end{matrix}\right]\overset{\text{Unif}(9)}\mapsto\left[\begin{matrix}\_&\_&\_\\\ \_&\_&\_\\\ \_&b&\_\\\end{matrix}\right]\overset{\text{Unif}(8)}\mapsto\left[\begin{matrix}\_&\_&\_\\\ \_&\_&b\\\ \_&b&\_\\\end{matrix}\right]$$ $$\overset{\text{Unif}(7)}\mapsto\left[\begin{matrix}\_&\_&\_\\\ a&\_&b\\\ \_&b&\_\\\end{matrix}\right]\overset{\text{Unif}(6)}\mapsto\left[\begin{matrix}\_&a&\_\\\ a&\_&b\\\ \_&b&\_\\\end{matrix}\right]\overset{\text{Unif}(5)}\mapsto\left[\begin{matrix}\_&a&\_\\\ a&a&b\\\ \_&b&\_\\\end{matrix}\right]$$
This question may be trivial but I've not worked with probability distributions on sets of matrices nor with comparing random processes.
