If (x=\frac{9!}{7!}) and (y=\frac{6!}{4!}), what is the value of (\frac{x}{y})?

Correct answer: (\frac{12}{5})

The direct answer is option A, \(\frac{12}{5}\). A factorial means multiplying a positive integer by every smaller positive integer: \(9!=9\times8\times7!\), so \(\frac{9!}{7!}=9\times8=72\). Similarly, \(6!=6\times5\times4!\), so \(\frac{6!}{4!}=6\times5=30\). Therefore, \(\frac{x}{y}=\frac{72}{30}=\frac{12}{5}\) after dividing numerator and denominator by 6. Option A is correct because it is the simplified ratio. Option B, \(\frac{5}{12}\), is the reciprocal and would represent \(\frac{y}{x}\), not \(\frac{x}{y}\). Option C, \(\frac{6}{5}\), comes from an incomplete or incorrect cancellation and does not equal \(72\div30\). Option D, \(\frac{5}{6}\), is also a reciprocal-type result and is not the required ratio. Remember: cancel factorials first, then divide x by y in the order given.