Using the fact that the Parikh matrix mapping is not an injective mapping, the paper investigates some properties of the set of the binary words having the same Parikh matrix; these words are called “amiable” Some results concerning the conditions when the equivalence classes of amiable words have more than one element, a characterization theorem concerning a graph associated to an equivalence class of amiable words, and some basic properties of a rank distance defined on these classes are the main subjects considered here.