Counting Paths
Now here is a favorite question type of mine. How many times have you see a question that starts with "A frog sitting at the point..." or "Sonya the frog chooses a point..." in math competitions? What's with the frog?
Anyway, I digress. The point is, counting number of paths between two points can be very hard problems, and there are many creative ways to do it, making it one of the most challenging and fun question types.
Let's start with the basics:
AMC 8 LEVEL PROBLEMS
Counting paths on a grid: refer to this video clip for a primer
Counting paths in a graph: What if the moves a more complicated (not restrained to just up or right any more), and there are more branches? This is a good strategy to use (in essence, it is just organized caseworks).
1986 AJHSME Problem 9
AIME LEVEL PROBLEMS