Factorials and the counting principle

Build the foundations of combinatorics with factorials and the fundamental counting principle. This precalculus extension practice moves from factorial notation and cancellation to staged choices, code restrictions, disjoint cases and unequal branches. Single-select and multiselect questions emphasize what is being counted and when addition or multiplication applies. Detailed explanations expose common errors such as treating factorials as linear expressions, admitting leading zeroes or assuming every branch has the same number of options. All factorial inputs are nonnegative integers; formal combinations and probability are reserved for later sets.