oi1. Find the number of positive integer solutions for the equation
$$
x+y+z=20
$$
Solving for $z$: $=1$, substituting into (1), we get $x+y=19$. For this boundary, when $x$ takes the values $1, 2, \cdots, 18$ $=2$, there are 17 sets of positive integer solutions. Continuing this way, when $z=18$, there is only 1 set of positive integer solutions. Therefore, the total number of positive integer solutions for equation (1) is
$$
18+17+\cdots+1=171 .
$$