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

Correct answer: \((n+2)(n+1)\)

Since \((n+2)!=(n+2)(n+1)!\), \(\frac{(n+2)!}{(n+1)!}=n+2\). Similarly, \((n+1)!=(n+1)n!\), so \(\frac{(n+1)!}{n!}=n+1\). Therefore, the complete expression is \((n+2)(n+1)\). The option \(n+2\) represents only the first ratio and misses the factor \(n+1\). Exam tip: In factorial ratios, expand the larger factorial up to the smaller one and cancel common factors.