When creating homework for my students, I came up with this: How many lists of length $5$ can be made from the set $ \{A,B,C,D,E,F,G,H,I\} $ if we cannot repeat a letter and they must be in alphabetical order?
Now, the way I would solve this would be to do a case by case analysis depending on the first letter. So, start with $A$ and count the lists by choosing a second letter and so on. Then do the same for $B$ the first letter. My question is whether there is a much less computationally long answer or an answer that is more instructive.