a: \(\left(\frac12-1\right)\left(\frac13-1\right)\cdot\ldots\cdot\left(\frac{1}{1999}-1\right)\)
\(=\frac{-1}{2}\cdot\frac{-2}{3}\cdot\ldots\cdot\frac{-1998}{1999}\)
\(=\frac12\cdot\frac23\cdot\ldots\cdot\frac{1998}{1999}=\frac{1}{1999}\)
b: \(\frac{5\cdot18-10\cdot27+15\cdot36}{10\cdot36-20\cdot54+30\cdot72}\)
\(=\frac{5\cdot9\cdot2-5\cdot9\cdot2\cdot3+5\cdot9\cdot3\cdot4}{10\cdot18\cdot2-10\cdot18\cdot2\cdot3+10\cdot18\cdot3\cdot4}\)
\(=\frac{5\cdot9\left(2-2\cdot3+3\cdot4\right)}{10\cdot18\cdot\left(2-2\cdot3+3\cdot4\right)}=\frac{5}{10}\cdot\frac{9}{18}=\frac12\cdot\frac12=\frac14\)
c: \(-1\frac12\left(-1\frac13\right)\cdot\ldots\cdot\left(-1\frac{1}{1999}\right)\)
\(=\frac{-3}{2}\cdot\frac{-4}{3}\cdot\ldots\cdot\frac{-2000}{1999}=\frac32\cdot\frac43\cdot\ldots\cdot\frac{2000}{1999}=\frac{2000}{2}\)
=1000
