Lessons from Letter Combinations

$1.$ The product rule tells you the size of the output before you generate it.

$2.$ Backtracking generates all combinations by making independent choices at each step.

$3.$ Exponential growth (like $4^n$) means the problem becomes infeasible for large inputs.

You now have the complete toolkit for counting: sum rule, product rule, DP recurrences, and backtracking generation.