If \( \frac{(2n)!}{(2n-4)!}=1680 \), what is the value of (n)?

Correct answer: 4

Expanding the factorial ratio gives \(\frac{(2n)!}{(2n-4)!}=(2n)(2n-1)(2n-2)(2n-3)\). The number 1680 can be written as the product of four consecutive integers: \(5\times6\times7\times8\). Hence \(2n=8\), so \(n=4\). For instance, if \(n=5\), the product would be \(10\times9\times8\times7\), not 1680. Exam tip: In factorial-ratio questions, cancel the common factorial part and write the remaining consecutive factors.