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

Correct answer: ((n+1)(n+3))

Factorial notation allows us to cancel common factorial factors by expanding only as far as needed. We use (n+2)!=(n+2)(n+1)n! and (n+1)!=(n+1)n!. This turns both fractions into ordinary algebraic terms. The simplification assumes the factorials are defined, normally for non-negative integer n.

The first fraction becomes (n+2)(n+1)n!/n!=(n+2)(n+1). The second becomes (n+1)n!/n!=n+1. Adding gives (n+1)(n+2)+(n+1). Taking the common factor n+1 gives (n+1)[(n+2)+1]=(n+1)(n+3). Therefore option A is correct. The unsimplified terms must not be added by combining their factorial numerators incorrectly.