The number of triplets $(x,y,z),$ where $x,y,z$ are distinct non negative integers satisfying $x+y+z=15,$ is:

$x+y+z=15$

Total no. Solution $={}^{15+3-1}C_2=\mathrm{136...}\left(1\right)$

Let $x=y\ne z$

$2x+z=15\Rightarrow z=15-2t\Rightarrow t\in \{0,1,2,\mathrm{.....7}\}-\left\{5\right\}$

$\therefore$ 7 solutions

$\therefore$ There are 21 solutions in which exactly

Two of x,y,z are equal ........(2)
There is one solution in which  $x=y=z$......(3)
Required answer $=136-21-1=114.$