If \( \frac{(n+3)!-(n+2)!}{(n+1)!}=121 \), what will be the value of (n)?

Correct answer: 9

Using \((n+3)!=(n+3)(n+2)!\), the numerator becomes \((n+2)![(n+3)-1]=(n+2)(n+2)!\). Hence the given expression simplifies to \(\frac{(n+2)(n+2)!}{(n+1)!}=(n+2)^2\). Thus, \((n+2)^2=121\), giving \(n+2=11\) and \(n=9\). The other algebraic value, \(n+2=-11\), gives \(n=-13\), for which the factorials in the question are not defined. Exam tip: In factorial expressions, first expand the larger factorial in terms of the smaller one.