I'm trying a few elementary counting problems, and a few are proving very difficult (for me). I have the answers and explanations, which I understand, so that's not the problem. I don't want to memorize answers. The problem is systematically analyzing these problems. My intuition is almost always wrong with counting, so I don't want to rely on any intutive explanations (adding to my frustration, most explanations use the intuitive approach). Here are some problems I found troubling: My telephone rings 12 times each week, the calls being randomly distributed among the 7 days. What is the probability that I get at least one call each day? If n balls are placed at random into n cells, find the probability that exactly one cell remains empty A closet contains n pairs of shoes. If 2r shoes are chosen at random (2r<n), what is the probability that there will be no matching pair in the sample? Given problems like these, how do you systematically work through the problem.