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

$\mathrm{x}+\mathrm{y}+\mathrm{z}=15$
Total number of non-negative integers  ${=}^{\mathrm{n}+\mathrm{r}-1}{\mathrm{C}}_{\mathrm{r}-1}$
                                                                      $\begin{array}{l}={}^{17}{\mathrm{C}}_2\\ =136\end{array}$
If $\mathrm{x}=\mathrm{y}=\mathrm{z}\Rightarrow \mathrm{x}=\mathrm{y}=\mathrm{z}=5\to 1\mathrm{solution}$
if $\mathrm{x}=\mathrm{y}\Rightarrow 2\mathrm{x}+\mathrm{z}=15$
x = 1, 2, 3, 4, 5, 6, 7
$\therefore$7 solutions
Similarly $\mathrm{y}=\mathrm{z}\& \mathrm{z}=\mathrm{x}$ also have 7 solutions each.
$\therefore$Total no.of solutions in which atleast two of them are equal = 22
$\therefore$ Numbers of distinct integer solutions $=136-22-114$