What is the value of (\frac{15!-12!}{12!})?

Correct answer: (2729)

The direct answer is option B, 2729. Use factorial cancellation instead of calculating enormous factorials. Divide each term in the numerator by \(12!\): \(\frac{15!-12!}{12!}=\frac{15!}{12!}-\frac{12!}{12!}\). Now \(\frac{15!}{12!}=15\times14\times13=2730\), while \(\frac{12!}{12!}=1\). Therefore the result is \(2730-1=2729\). Option B is correct. Option A, 2728, is one less than the correct value and can result from an arithmetic error. Option C, 2730, is only the value of \(\frac{15!}{12!}\); it forgets the subtraction of 1. Option D, 2731, is too large and does not follow from the subtraction. The useful rule is \(\frac{m!}{r!}=m(m-1)\cdots(r+1)\) when \(m>r\), and a factorial divided by itself equals 1.