We study the question how difficult it is to compute the multi-homogeneous
B\'ezout number for a polynomial system of given number $n$ of variables
and given support $A$ of monomials with non-zero coefficients.
We show that this number is NP-hard to compute. It cannot even be efficiently
approximated within an arbitrary, fixed factor unless P = NP.
This is joint work with Gregorio Malajovich.