Suppose we have the NxN matrix A. Consider A as described below for N=8. A subset S of matrix A is given by the "x" in the matrix A. Note that this subset has as elements exactly k=2 "x"s per column and row of the original matrix A.

   1 2 3 4 5 6 7 8
1  x o o o o o o x
2  x x o o o o o o
3  o x x o o o o o
4  o o x x o o o o
5  o o o x x o o o 
6  o o o o x x o o
7  o o o o o x x o
8  o o o o o o x x

Suppose we randomly select m "x"s from the matrix with replacement. Then, we create an induced matrix with just the rows and columns corresponding to the selected "x"s. For instance, if the uniquely selected "x"s are (1,1),(3,2),(3,3),(6,6),(8,8) then the resultant matrix is

   1 2 3 6 8
1  x o o o x
3  o x x o o
6  o o o x o
8  o o o o x

Let X be the number of "x"s in the resultant matrix. For the given example, this value is 6.

My Question: How we can calculate the average number of "x"s in the resultant matrix for a given m (E[X](N,k,m))?

I have already calculated the average number of columns E(C) when m "x"s are randomly selected with replacement. Note that for this case E(C) is equal to the average number of rows E(R) and can be calculated as:

E[C](N,m)=N*P_chosen, where P_chosen=1-(1-(1/N))^m) is the probability that a column of the original matrix is selected at least once.

Thank you for any help!