12. Given that $x$, $y$, and $z$ are all positive integers. Then the equation $x+y+z=$ 15 has solutions.
12. 91 .

Express 15 as 15 ones, i.e., $1+1+\cdots+1=15$, where any 2 of the 14 plus signs are chosen, and the 1s separated by these two plus signs are combined into a single number to form a solution of the equation. Therefore, the number of solutions is $\mathrm{C}_{14}^{2}=91$.