  The baryon mass operator is studied within a combined expansion in $1/N_c$
and perturbative $SU(3)$ flavor symmetry breaking, where $N_c$ denotes the
number of quark charges. Flavor projection operators are used to classify the
baryon operators involved in the expansion, which fall into the flavor
representations $1$, $8$, $10+\overline{10}$, $27$, $35+\overline{35}$ and
$64$. This approach allows one to incorporate up to third-order flavor symmetry
breaking in the baryon mass operator in a rigorous and systematic way. Previous
work on the subject is considered to validate the approach. A fit to data is
performed to evaluate the free parameters in the theory and to produce some
numerical values of baryon masses. Results are consistent and reaffirm the
striking success of the $1/N_c$ expansion.
