  Let S be a finitely generated standard multi-graded algebra over a Noetherian
local ring A. This paper first expresses mixed multiplicities of S in term of
Hilbert-Samuel multiplicity that explained the mixed multiplicities S as the
Hilbert-Samuel multiplicities for quotient modules of S. As an application, we
get formulas for the mixed multiplicities of ideals that covers the main result
of Trung-Verma 2007 [TV].
