What is the difference between ( \frac{20!}{18!\cdot2!} ) and ( \frac{19!}{17!\cdot2!} )?

Correct answer: (19)

First simplify each factorial quotient separately. For the first term, \(\frac{20!}{18!\,2!}=\frac{20\times19\times18!}{18!\times2}=\frac{20\times19}{2}=190\). For the second, \(\frac{19!}{17!\,2!}=\frac{19\times18\times17!}{17!\times2}=\frac{19\times18}{2}=171\). The common factorial parts cancel before any subtraction is done.

The question asks for the difference, so subtract the second value from the first: \(190-171=19\). Equivalently, the expressions are consecutive combination values \(\binom{20}{2}\) and \(\binom{19}{2}\), whose difference is 19. Therefore option B is correct. Options 18 and 20 can result from subtracting nearby factors incorrectly, while 21 does not follow from the calculation.