Let $\vec a=\hat i+\hat j+\hat k$ and let $\vec r$ be a variable vector such that $\vec r.\hat i,\,\vec r.\hat j$ and $\vec r.\hat k$ are positive integers. If $\vec r.\vec a≤12$ then the total number of such vectors is:

${^{12}C}_3$

$\vec a=\hat i+\hat j+\hat k$; $\vec r.\hat i,\,\vec r.\hat j,\,\vec r.\hat k$ are positive integers.

Let $\vec r=l\hat i+m\hat j+n\hat k$ where l, m, n > 0 ⇒ $\vec r.\vec a= l+m+n≤12$

$⇒l+m+n≤12$

Thus number of positive integral solution is ${^{12-1}C}_{3-1}+{^{11-1}C}_{3-1}+...+{^{4-1}C}_{3-1}$

$={^{12}C}_3$