Three numbers are chosen at random from the first $20$ numbers$1, 2, 3, \dots 20.$ The probability the selected
numbers forms a $G.P.$, is

Three numbers can be chosen out of $20$ in ${ }^{20}{C}_3$

ways.
For different values of common ratios, $G.P.$’s are
listed below.

$\therefore$ probability of required event $=\frac{11}{{ }^{20}{C}_3}$