You asked a really interesting question here! The Day 3: Challenge (Part 1) question explanation jumps a little bit ahead to show you what tricks you will see later on. To answer your question in a nutshell, both methods work equally well for simple problems, but maybe not for more complicated ones. Professor Loh likes to choose problems that are doable by what we call "brute force," so everyone has a chance to attack them. However, if you asked a different question that was on a larger scale, like how many 10-digit numbers you can make using the digits 1, 2 and 3, there would be so many ways to list that you couldn't list them all easily. In this case, alternative counting tricks are much more efficient to use!
This relates to math in general: there are often many different ways of doing a problem. Once you have solved a problem with one technique, there is a huge pleasure that comes from seeing that there was another, perhaps very different, way to solve the same problem. Almost all of the problems in this course have this characteristic. It's not just a fun way to learn math, it stretches your mind in a good way!
The Daily Challenge Team