What is the value of (\frac{18!}{16!\times 2!} - \frac{17!}{15!\times 2!})?

Correct answer: (17)

Factorial notation allows common factors to be cancelled before doing the arithmetic. Since 18! = 18×17×16! and 17! = 17×16×15!, the factorial fractions become products of only the required consecutive integers. This makes the subtraction simple and avoids calculating the very large factorials themselves.

The first fraction is \(\frac{18!}{16!\times2!}=\frac{18\times17}{2}=153\). The second is \(\frac{17!}{15!\times2!}=\frac{17\times16}{2}=136\). Their difference is \(153-136=17\). Equivalently, factor out 17 and use \(\frac{17(18-16)}{2}=17\). Thus option B is correct.