Philonumberist

$x + y + 3z = 27\\ x + y = 27 - 3z$

Since $x, y$ and $z$ are positive integers, the only permissible values of $z = \{1,2,3,\cdots,8\}$
When $z = 1$ , $x + y = 24$ and the number of ordered pairs $(x, y)$ satisfying this equation = $^{24-1}C_{2-1} = ^{23}C_1$ .
When $z = 2$ , $x + y = 21$ and the number of ordered pairs $(x, y)$ satisfying this equation = $^{21-1}C_{2-1} = ^{20}C_1$ .
When $z = 3$ , $x + y = 18$ and the number of ordered pairs $(x, y)$ satisfying this equation = $^{18-1}C_{2-1} = ^{17}C_1$ . $\vdots$
When $z = 8$ , $x + y = 3$ and the number of ordered pairs $(x, y)$ satisfying this equation = $^{3-1}C_{2-1} = ^{2}C_1$ .

So, the total number of waying of finding the ordered triplets $\begin{aligned} (x, y, z) & = ^{23}C_1 + ^{20}C_1 + ^{17}C_1 + \cdots + ^{2}C_1\\ & = 23 + 20 + 17 + \cdots + 2\\ & = \color{#3D99F6}{\boxed{100}} \end{aligned}$