The dominoes scheme is defined as a scheme of random tiling with poliomino tiles with $r$ ends and $n$ figures from 0 to $(n-1)$ on the ends of tiles of all possible compositions with repetitions, regardless of their order. This scheme is studied, together with an analogous one, with a fixed minimal figure $rge m$ at the outcome of a random choice of a tile from a complete domino set, using the following ways of enumerative combinatorics: design of the enumeration procedure for the numbered outcomes of the scheme, determination of their number, solution of the corresponding numeration problem (i.e. establishment of one-to-one correspondence between numbers and types of the outcomes of the scheme), finding of their probabilistic distribution and modeling of their possible outcomes