  This paper defines the Euler-Poincar\'{e} characteristic of joint reductions
of ideals which concerns the maximal terms in the Hilbert polynomial;
characterizes the positivity of mixed multiplicities in terms of minimal joint
reductions; proves the additivity and other elementary properties for mixed
multiplicities. The results of the paper together with the results of [17] seem
to show a natural and nice picture of mixed multiplicities of maximal degrees.
