Nothing in this chapter is new mathematics. Every idea here — the product and sum rules, permutations and combinations, Pascal's triangle, recurrence relations, graphs, and trees — was taught in Discrete Mathematics I. What's new is the question this course is about to ask of that toolkit: not "how many," but "which counting argument, and why does it work."
This chapter refreshes each tool quickly, in exactly the framing this course will use it, so Chapter 1 can start immediately from a shared foundation instead of pausing to relearn material mid-argument. The last two sections are the only genuinely new material here — manipulating a finite sum by shifting its index or splitting it into pieces, and turning that manipulation into a closed form. Both get leaned on the moment Chapter 2 starts summing counts across cases.
By the end, every section from here through the end of the course will be able to assume this toolkit is already in your hands.