On Catalan numbers

This paper provides problems on generating Catalan numbers and some of its applications for readers to be familiar with the sequence. The articles entitled Catalan Strikes Again! How Likely Is A Function To Be Convex? written by Roger B. Eggleton and Richard K. Guy and Mathematical Games by Martin Gardner were the bases for this paper. The concepts used are basically related to such fields as Combinatorics, Probabilities and College Algebra. The problem Is the probability of the function (0, fo), (1, f1),... (n, fn) with n+1 points and within the interval [0,1] convex? was considered to generate the formula cn/(n!)2