What is the simplified form of \( \frac{(n+4)!}{(n-1)!(n+3)(n+2)} \)?

Correct answer: \(n(n+1)(n+4)\)

Expand \((n+4)!\) down to \((n-1)!\): \((n+4)!=(n+4)(n+3)(n+2)(n+1)n(n-1)!\). In the given expression, \((n-1)!\), \((n+3)\), and \((n+2)\) cancel, leaving \(n(n+1)(n+4)\). Hence, option A is correct. Option B incorrectly keeps \((n+2)\), although it cancels with the denominator. Exam tip: in factorial fractions, expand the larger factorial only up to the smaller factorial before cancelling.