If \(\frac{(n+4)!}{(n+1)!}=504\), what is the value of (n)?

Correct answer: 5

Cancelling \((n+1)!\) after expanding \((n+4)!\) gives \(\frac{(n+4)!}{(n+1)!}=(n+4)(n+3)(n+2)\). Hence, \((n+4)(n+3)(n+2)=504\). Since \(7\times8\times9=504\), we get \(n+2=7\), so \(n=5\). For example, \(n=4\) gives \(6\times7\times8=336\), not \(504\). Exam tip: In factorial fractions, expand the larger factorial only up to the smaller factorial and cancel common terms.