jmc

We have that $$\begin{array}{rl}\frac{(1+17)\left(1+\frac{17}{2}\right)\left(1+\frac{17}{3}\right)\cdots \left(1+\frac{17}{19}\right)}{(1+19)\left(1+\frac{19}{2}\right)\left(1+\frac{19}{3}\right)\cdots \left(1+\frac{19}{17}\right)}& =\frac{\frac{18}{1}\cdot \frac{19}{2}\cdot \frac{20}{3}\cdots \frac{36}{19}}{\frac{20}{1}\cdot \frac{21}{2}\cdot \frac{22}{3}\cdots \frac{36}{17}}\\ & =\frac{\frac{36!/17!}{19!}}{\frac{36!/19!}{17!}}\\ & =\boxed{1}.\end{array}$$