In this paper, we find configurations of points in $n$-dimensional projective space $\left( {{\mathbb{P}}^{n}} \right)$ which simultaneously generalize both $k$-configurations and reduced 0-dimensional complete intersections. Recall that $k$-configurations in ${{\mathbb{P}}^{2}}$ are disjoint unions of distinct points on lines and in ${{\mathbb{P}}^{n}}$ are inductively disjoint unions of $k$-configurations on hyperplanes, subject to certain conditions. Furthermore, the Hilbert function of a $k$-configuration is determined from those of the smaller $k$-configurations. We call our generalized constructions ${{k}_{D}}$-configurations, where $D\,=\,\left\{ {{d}_{1}},\ldots ,{{d}_{r}} \right\}$ (a set of $r$ positive integers with repetition allowed) is the type of a given complete intersection in ${{\mathbb{P}}^{n}}$. We show that the Hilbert function of any ${{k}_{D}}$-configuration can be obtained from those of smaller ${{k}_{D}}$-configurations. We then provide applications of this result in two different directions, both of which are motivated by corresponding results about $k$-configurations.