Teaching the art of counting can be quite difficult. Many undergraduate students have difficulty separating the ideas of permutation, combination, repetition, etc. This article develops some examples to help explain some of the underlying theory while looking carefully at the selection of various subsets of objects from a larger collection. The author's main purpose is to examine a way of extending some basic counting techniques in order to generate some simple projects for sophomore or junior level discrete mathematics students. The ideas developed here would be appropriate for students who have learned the basic counting techniques and are ready to look at some more challenging problems. Advanced students from a senior level combinatorics class would also benefit from working through these examples.