  It is shown in our earlier paper that, using only tools of elementary
geometry, the classical Routh's theorem for triangles can be fully extended to
tetrahedra. In this article we first give another proof of Routh's theorem for
tetrahedra where methods of elementary geometry are combined with the
inclusion-exclusion principle. Then we generalize this approach to $(n-1)-$
dimensional simplices. A comparison with the formula obtained using vector
analysis yields an interesting algebraic identity.
