If \( \frac{(n+6)!}{(n+2)!}-\frac{(n+5)!}{(n+1)!}=2880 \), what is the value of (n)?

Correct answer: 5

\(\frac{(n+6)!}{(n+2)!}=(n+3)(n+4)(n+5)(n+6)\) and \(\frac{(n+5)!}{(n+1)!}=(n+2)(n+3)(n+4)(n+5)\). Taking the common factors, their difference becomes \(4(n+3)(n+4)(n+5)\). Hence \(4(n+3)(n+4)(n+5)=2880\), so \((n+3)(n+4)(n+5)=720\). Since \(8\times9\times10=720\), \(n=5\). For the nearby option 6, the product would be \(9\times10\times11\), not 720. Exam tip: first expand factorial ratios into consecutive factors and cancel common terms carefully.