question_answer

Let \[\overrightarrow{a}\text{ =}\hat{i}+\hat{j}+\hat{k}\] and \[\overrightarrow{r}\] be a variable vector such that \[\overrightarrow{r}.\,\,\hat{i},\]\[\overrightarrow{r}.\,\,\hat{j},\] and \[\overrightarrow{r}.\,\,\hat{k}\] are positive integers. If \[\overrightarrow{r}.\,\,\overrightarrow{a}\le 12,\] then the total number of such vectors is _________.

A)  \[{{12}_{{{c}_{4}}}}\]                                   

B)  \[{{8}_{{{c}_{2}}}}\]

C)  \[{{9}_{{{c}_{3}}}}\]                         

D)  \[{{12}_{{{c}_{3}}}}\]

E)  None of these