If \( \frac{(n+7)!}{(n+4)!}-\frac{(n+6)!}{(n+3)!}=720 \), what is the value of (n)?

Correct answer: 10

\(\frac{(n+7)!}{(n+4)!}=(n+7)(n+6)(n+5)\), while \(\frac{(n+6)!}{(n+3)!}=(n+6)(n+5)(n+4)\). Factoring out the common terms gives the difference as \((n+6)(n+5)[(n+7)-(n+4)]=3(n+6)(n+5)\). Thus, \(3(n+6)(n+5)=720\), so \((n+6)(n+5)=240\). The consecutive numbers 15 and 16 have product 240, hence \(n+5=15\) and \(n=10\). For option 9, the product would be \(14\times15=210\), so it is not correct. Exam tip: Expand factorial ratios into consecutive factors before simplifying common terms.