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

Correct answer: (2n+3)

The useful factorial rule is \(m!=m(m-1)!\). Apply it separately to each fraction rather than combining the two fractions at the beginning. In the first fraction, \((n+2)!=(n+2)(n+1)!\), so division by \((n+1)!\) leaves \(n+2\). In the second, \((n+1)!=(n+1)n!\), so division by \(n!\) leaves \(n+1\).

The original expression therefore becomes \((n+2)+(n+1)\). Combining like terms gives \(n+n+2+1=2n+3\). Thus option C is correct. The answer is not \(2n+2\), because the constant terms 2 and 1 add to 3. The simplification assumes the factorials are defined, as usual in this context, for the relevant nonnegative integer values of \(n\).