We consider the three-particle Schrödinger operator \(H_{a,b},\) \(a,b>0,\) associated with a system of three particles (two bosons with mass 1, interacting via zero-range pair potentials of strength \(b>0,\) and one arbitrary particle with mass \(m>0\) , interacting with bosons via a zero-range potential of strength \(a>0\) ) on the three-dimensional lattice \( \mathbb {Z}^3.\) We establish the exact number of the discrete eigenvalues of \(H_{a,b}\) below its essential spectrum in the cases when a boson interacts with the third particle strongly (that is b is fixed and a is large) and when two bosons strongly interact (that is for fixed \(a>0\) and large b).
