number of positive solution

From $2x+y+z = 15$ , $\implies z = 15-2x-y$ . This means that there is only one $z$ for one $x$ and one $y$ . To find the total number $N$ of solutions we need only find the total number possible $(x,y)$ pairs. It is obvious that $x$ ranges from $1$ to $6$ . And for a value for $x$ , the possible number of $y$ 's is $n(x) = 15-2x - 1 = 14-2x$ . Therefore, the total number of solutions is:

$\begin{aligned} N & = \sum_{x=1}^6 (14-2x) = 2 \sum_{x=1}^6 (7-x) = 2 \sum_{k=1}^6 k = 2 \times \frac {6(6+1)}2 = \boxed{42} \end{aligned}$