Let In be the semigroup of all full transformations on Xn = {1, 2, 3,..., n}. A transformation a on Xo is a contraction if Ita – ya < 1-y (Vr,y e Xn) The full contraction transformations semigroup CT, is a sub-semigroup of all contractions on Xm. In this thesis, I will deduce some results on some equivalence classes defined on the set of idempotents of the full contraction transformations semigroup; E(CTn). Then, I will investigate some algebraic properties of CT such as reg ularity and some Green's Relations. Finally, I will study combinatorial proper ties of CT, using some combinatorial functions: • F(n, p,m) counting transformations with a fixed image size and a fixed number of fixed points. • Fn, p, k) counting transformations with a fixed image size and a fixed maximum for the image.