Which is the correct expansion of (n + 3)!/(n + 1)!?

Correct answer: (n + 3)(n + 2)

The governing concept is factorial cancellation. Expand the numerator only until the denominator factorial appears: (n + 3)! = (n + 3)(n + 2)(n + 1)!. Therefore, [(n + 3)(n + 2)(n + 1)!]/(n + 1)! simplifies by cancelling the common factor (n + 1)!, leaving (n + 3)(n + 2). Thus option A is correct. Option B incorrectly uses n + 1 as a separate factor instead of recognizing that the entire (n + 1)! cancels. Option C omits the factor n + 3, and option D retains the denominator factorial's factors unnecessarily. This method is valid wherever the factorials are defined, normally for nonnegative integer arguments.