  This paper defines mixed multiplicity systems; the Euler-Poincare
characteristic and the mixed multiplicity symbol of $\mathbb{N}^d$-graded
modules with respect to a mixed multiplicity system, and proves that the
Euler-Poincare characteristic and the mixed multiplicity symbol of any mixed
multiplicity system of the type $(k_1,\ldots,k_d)$ and the
$(k_1,\ldots,k_d)$-difference of the Hilbert polynomial are the same. As an
application, we get results for mixed multiplicities.
