I'm trying to tackle the following question:
Find the number of $m\times n$ matrices of $0$'s and $1$'s with mutually distinct rows.
I thought about inclusion-exclusion, but it gets me nowhere. Also, I'm afraid that I don't understand the term "mutually distinct rows"? does it mean that each row of the matrix is different?
Any help is appreciated, thanks!
As far as I understand "mutually distinct rows" means that the rows are pairwise distinct. Thus if you choose the rows one by one, there are $2^m$ choices for the first row (all elements of $\{0,1\}^m$), $2^m-1$ choices for the second row (all elements of $\{0,1\}^m$ except the first row), etc.
Hence in overall it gives $2^m (2^m-1) \ldots (2^m-n+1)$ such matrices.