What is the value of ( \frac{12!}{8!\cdot4!}-\frac{11!}{7!\cdot4!} )?

Correct answer: (165)

Each fraction should be simplified separately before subtracting. A factorial quotient of the form \(\frac{n!}{r!(n-r)!}\) represents a binomial coefficient, but direct cancellation is also clear here. The first term gives 495 and the second gives 330. Their difference is therefore 165, which is option A.

For the first term, \(\frac{12!}{8!4!}=\frac{12\times11\times10\times9}{4\times3\times2\times1}=495\). For the second, \(\frac{11!}{7!4!}=\frac{11\times10\times9\times8}{4\times3\times2\times1}=330\). Subtracting gives \(495-330=165\). Hence option A is correct. Subtracting factorial symbols before evaluating would be an invalid method.