# Are these random matrix processes equivalent?

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.

They are the same. That is, these two methods give induce the same probability distributions. One way to help see this is to note that the product of the number of choices ($$9\times8\times7\times6\times5$$) is the same in both cases.