A pair of pants $sum(0,;3)$ is a building block of oriented surfaces. The purpose of this paper is to determine the discrete conditions for the holonomy group $pi$ of hyperbolic structure of a pair of pants. For this goal, we classify the relations between the locations of principal lines and entries of hyperbolic matrices in $mathbf{PSL}(2,;mathbb{R})$. In the level of the matrix group $mathbf{SL}(2,;mathbb{R})$, we will show that the signs of traces of hyperbolic elements playa very important role to determine the discreteness of holonomy group of a pair of pants.