We present here by 'Preface': In this short course, we present the ways of counting in a systematic way so that students can develop and acquire all the methods for counting the number of cases with confidence by clearly distinguishing each case. In the U.S., students usually encounter such chapters as Permutations and Combinations in their mathematics class, but methods for all the possible cases will be rarely offered. When they encounter complicated cases and found themselves at a loss on what to do, they come to realize the fact that they have only known rather basic concepts of Permutations and Combinations. We hope that this short course will bring clear understandings on how to count all the cases in various situations. At first, we describe the summary as concisely as possible together with typical examples. Afterwards, you will make practice by engaging in solving problems in the following exercise sections. Those exercise sections will be divided according to the level of difficulty or whether they contain combined concepts. After finishing the course, we recommend that you engage in solving illustrated problems periodically so that you refresh your understandings on the subject.