In schemes S for placing r particles in n distinguishable cells, their placement in selected m cells – schemes S∗ is studied, for which the analysis is carried out by a new enumerative method (EM) in extended areas of enumerative combinatorics: finding the number of outcomes and, based on the construction of a model of their non-repetitive numbered enumeration, solving the numbering problem for them in direct and inverse statements to establish a one-to-one correspondence between their types and numbers, and introducing a controlled probabilistic distribution on the set of outcomes of the scheme.