If $A=\left[\begin{array}{ccc}a& x& y\\ x& b& z\\ y& z& c\end{array}\right]$ where $a,b,c,x,y,z\in \{1,2,3,4,5,6\}$ and also $a,b,c,x,y,z$ are distinct, then the number of matrices as $A$ with trace equal to 10 is 

$\because T_r\left(A\right)=a+b+c=10$

Therefore it may be $(1,3,6),(1,4,5).(2,3,5)$

$\therefore$ number of matrices $=3!\times 3\times 3!$

Arranging diagonal arranging non-diagonal elements $=3{(3!)}^2$