Enumerative Combinatorics is dealing with enumeration of finite sets. Considering a certain property of combinatorial objects, we would like to know how many objects have this property. In ideal case, we would like to know it for objects of any size n. Moreover, the full description of the considered objects would include a generating function, different recurrence and differential equations, asymptotics, and averages. Some parts of these are closely related to probability distributions, which is understandable, because enumerating forms sequences of nonnegative numbers. Despite the simple description and obvious goals, solutions of many of the problems of Enumerative Combinatorics are hard or incomplete. The Experimental Mathematics turns out to be an essential part of solutions of different mathematical problems. Today it appears in many areas beyond the numerical analysis. In this project we shall consider some problems arising in Enumerative Combinatorics, and use both, theoretical and experimental, approaches to analyze the existing results, to make conjectures, and to prove new results.