  Using standard group theoretical techniques we construct the exact wave
function of the $[{\bf 70},1^-]$ multiplet in the orbital, spin and flavor
space. This symmetric wave function is compared to that customarily used in the
$1/N_c$ expansion, which is asymmetric. The comparison is made by analyzing the
matrix elements of various operators entering the mass formula. These matrix
elements are calculated by the help of isoscalar factors of the permutation
group, specially derived for this purpose as a function of $N_c$. We also
compare two distinct methods used in the study of the $[{\bf 70},1^-]$
multiplet. In the first method the generators are divided into two parts, one
part acting on a subsystem of $N_c-1$ quarks called core and another on the
separated quark. In the second method the system is treated as a whole. We show
that the latter is simpler and allows to clearly reveal the physically
important operators in the mass formula.
