The basic physical processes responsible for X-ray emission from thin plasmas are considered. Collisional ionization or photoionization of inner shells of neutral atoms and ions leads to the creation of a vacancy in one of the inner shells of the ion or atom, which is filled by a cascade of radiative (fluorescent) and nonradiative (Auger) transitions. The net result is the ejection of several electrons and photons, leaving the atom in a multiply ionized state. In this paper, the distribution of the number of emitted photons and electrons after the creation of a hole in an inner shell of an atom or ion is calculated for all ions from H to Zn. The method consists of two stages: the calculation of transition rates for a given electron configuration, and calculation of probabilities of the several cascade sequences using these transition rates.