Measurements of K x-ray or Auger satellite (and hypersatellite) intensities provide information on KL/sup v/ (and K/sup 2/L/sup v/) hole production in close collisions of ions with atoms. The L-shell vacancy distributions usually are nearly binomial. The binomial distribution follows immediately from a statistical model in which the creations of the various holes are regarded as mutually independent. This distribution is specified by a single parameter, p, the mean L-shell vacancy probability per electron, which can depend on the speed v, the nuclear charge Z and the ionic charge q of the projectile, and on the nuclear charge of the target. Almost all calculations of p, aside from our work, have been limited to first-order collision approximations. Moreover, all such calculations have included only impact ionization and, consequently, have neglected electron transfer to the projectile. The first-order calculations have had enough success for proton and alpha-particle impact to remain in vogue for a decade, but they predict that p is proportional to Z/sup 2/, whereas anti p must remain less than or equal to 1, which requires a unitary collision approximation. We have obtained the saturation of p with Z in two unitary schemes, the first Magnus approximation and the more refined coupled-channel approximation. In both schemes we used a single-centered expansion (SCE) in spin-orbitals representing bound and unbound states of the target. Typically, we employ ten radial functions for each of four angular momenta (s, p, d, and f states).