Generating random 0/1 matrix with fixed features in R

I want to generate a random M x N matrix with zeros and ones with the following special properties:

1) The ones are only in m of the M rows.

2) The ones are only in n of the N columns.

Suppose I am ONLY given that M=10, N=10, m=6 and n=4. One possible random matrix is given by

      [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
 [1,]    0    0    1    0    0    0    1    0    1     0
 [2,]    0    0    1    0    0    0    1    0    1     0
 [3,]    0    0    0    0    0    0    0    0    0     0
 [4,]    0    0    0    0    0    0    0    0    0     0
 [5,]    0    0    1    0    0    0    1    0    0     0
 [6,]    0    1    0    0    0    0    0    0    0     0
 [7,]    0    0    0    0    0    0    0    0    0     0
 [8,]    0    0    1    0    0    0    0    0    0     0
 [9,]    0    0    1    0    0    0    0    0    1     0
[10,]    0    0    0    0    0    0    0    0    0     0

For reproducibility, I've artificially generated the above "random" matrix using

ex <- matrix(0,10,10)
ex[1,3] <- ex[1,7] <- ex[1,9] <- ex[2,3] <- ex[2,7] <- 
  ex[2,9] <- ex[5,3] <- ex[5,7] <- ex[6,2] <- ex[8,3] <- ex[9,3] <- ex[9,9] <- 1

Note that

sum(rowSums(ex)>0)
[1] 6
sum(colSums(ex)>0)
[1] 4

which match exactly m and n above.

Question I can generate this in a brute force manner, sampling over the rows and columns, but I need to do this over thousands of such matrices (because the m and n will be different each time) and these matrices are large (M=5000 and N=8000, typically). Is there a way to do this efficiently in R?