In this series of lectures, we will start with some basictheory, examples and techniques of studying occurences of patterns,and waiting times in sequences of independent Bernoulli or multinomialtrials. We will discuss some exact theory, and asymptotics, such ascentral limit theorems and Poisson approximations. We will then proceed tospecial types of coincidences, such as matching, and the problem oflook-alikes. We also hope to present some of the modern developments onlongest increasing subsequences, and applications, such as to genetics.