What is the simplified form of \( \frac{(n+6)!}{(n+4)!}+\frac{(n+5)!}{(n+3)!} \)?

Correct answer: \(2(n+5)^2\)

\(\frac{(n+6)!}{(n+4)!}=(n+6)(n+5)\) and \(\frac{(n+5)!}{(n+3)!}=(n+5)(n+4)\). Hence, the sum is \((n+5)[(n+6)+(n+4)]\). This gives \((n+5)(2n+10)=2(n+5)^2\), so option A is correct. Option B has \(2n+9\), whereas the correct sum inside the bracket is \(2n+10\). Exam tip: In factorial division, cancel through the denominator factorial and multiply the remaining consecutive factors.