How many different binary matrices are there of size n x m, where rows can be interchanged and still remain the same matrix?

Question

I think it's the same as finding the number of equivalent matrices by elementary Gaussian transformations, since it's binary.

Maybe the question could be rephrased to the number of linearly independent vectors in a binary matrix and therefore they would be the total of different matrices echeloned.

I don't get the combinatorics

Example

For 2 x 2 matrices. I have 16 different binary matrices, but the matrix {01,10} is equal to {10,01} if I change the rows and I would only have 10 different matrices.

I would like to know if it is possible to find a formula for matrices of n x m

Answer 1

There are $2^m$ possible choices for a row, and what we are interested in (if I understand the question correctly) is the number of multisets of size $n$ composed of those rows. The number of such multisets is

$$\binom{2^m+n-1}{n}$$ by a well-known formula.