This paper defines mixed multiplicity systems; the Euler-Poincaré characteristic and the mixed multiplicity symbol of N^d-graded modules with respect to a mixed multiplicity system, and proves that the Euler-Poincaré characteristic and the mixed multiplicity symbol of any mixed multiplicity system of the type (k₁,..., k_d) and the (k₁,..., k_d)-difference of the Hilbert polynomial are the same. As an application, we obtain results for mixed multiplicities.