What is the value of \( \frac{15!}{12!\cdot3!}\times\frac{3}{13} \)?

Correct answer: (105)

The expression uses factorial cancellation and then a simple multiplication. A factorial such as \(15!\) contains all factors from 1 through 15, while \(12!\) contains those from 1 through 12. Dividing them leaves only \(13\times14\times15\). The factor \(3!\) must also be included in the denominator, so the first part is \(\frac{13\times14\times15}{6}=455\). This makes the calculation shorter and less error-prone.

Now multiply by \(\frac{3}{13}\). Cancelling 13 first gives \(\frac{14\times15\times3}{6}=14\times15\times\frac12=105\). Therefore the value is 105, which is option A. The other numerical options result from incomplete cancellation or incorrect factorial arithmetic.