Structures for the metal-centered 10-vertex pnictogen clusters M@Pn(10)(4+) (M = Ni, Pd, Pt; Pn = As, Sb, Bi) based on polyhedra with 3-fold, 4-fold, and 5-fold symmetry have been studied by density functional theory. Among these nine M/Pn combinations, only Pd@Bi-10(4+) and Pt@Bi-10(4+) are predicted to have the D-5d pentagonal antiprism as the lowest energy structure in accord with experimental observation of this cluster in the ternary halide Bi14PdBr10 as well as the prediction of the Wade-Mingos rules for these arachno systems. The lowest energy structures for the arsenic and antimony clusters M@Pn(10)(4+) (Pn = As, Sb) and Ni@Bi-10(4+) are predicted to have structures derived from a tetracapped trigonal prism that has been severely distorted for M@As-10(4+) (M = Pd, Pt). The volumes of the As-10 polyhedra other than the pentagonal prism are too small to contain interstitial palladium or platinum atoms so that major distortions are predicted for such clusters leading to partial opening of the polyhedron.