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

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

Using factorial notation, \(n!=n(n-1)!\), so \(\frac{n!}{(n-1)!}=n\). Likewise, \((n+1)!=(n+1)n!\), so \(\frac{(n+1)!}{n!}=n+1\). Hence the full expression is \(n\times(n+1)=n(n+1)\). \(n^2\) would result only if both factors were \(n\). Exam tip: In factorial ratios, expand the larger factorial just enough to cancel the smaller one.