There is a very large body of literature in which the Nambu\char21{}Jona-Lasino (NJL) model has been applied in the study of mesons, baryons, and hadronic matter. One area of interest has been the calculation of corrections of order ${1/N}_{c}$ to various quantities calculated in the leading (Hartree) approximation. Of particular interest is the work of Dmitra\ifmmode \check{s}\else \v{s}\fi{}inovi\ifmmode \acute{c}\else \'{c}\fi{}, Schulze, Tegen, and Lemmer [Ann. Phys. (N.Y.) 238, 332 (1995)]. These authors have considered the SU(2)-flavor NJL model and identified a set of diagrams, whose calculation yields ${1/N}_{c}$ corrections, while at the same time maintaining the relations, such as the Goldberger-Treiman relation, that follow from the chiral symmetry of the theory. In the present work we extend the work of Dmitra\ifmmode \check{s}\else \v{s}\fi{}inovi\ifmmode \acute{c}\else \'{c}\fi{} et al. to the case of SU(3)-flavor symmetry. In particular, we consider ${1/N}_{c}$ corrections to the quark vacuum condensates and to the ``gap equation.'' While ${1/N}_{c}$ corrections to the pion decay constant are significant, the corrections to the condensates are found to be quite small, as was the case in the SU(2)-flavor analysis. The Pauli-Villars regularization procedure is thought to be particularly useful for such calculations, since that procedure is known to maintain the symmetries of the theory. As part of our analysis we extend the Pauli-Villars regularization procedure to the case in which a particular diagram contains quarks of different constituent mass. Such diagrams appear when we generalize the SU(2)-flavor analysis to SU(3). We also show that, if we use the ``textbook'' definition of the Pauli-Villars method, in which fictitious particles of large mass are added to the theory, the resulting formalism may not be used in the case of the NJL model. What is done in practice is that particular divergent (and nondivergent) integrals are regulated in a fashion that yields results that are quite similar to the results of the covariant regularization procedure that is used by many authors. For constituent quark mass values of the magnitude usually used in the case of the SU(3)-flavor NJL model, we find that the ${1/N}_{c}$ corrections in the calculation of the pion decay constant play a role in obtaining satisfactory values for the quark vacuum condensates and the strength of the 't Hooft interaction.