Let $\vec a =\hat i +\hat j+\hat k$ and let $\vec r$ be a variable vector such that 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

${^{12}C}_3$

Let $\vec r=x\hat i+y\hat j+z\hat k$, where x, y, z are integers.

We have,

$\vec r.\hat i>0, \vec r.\hat j>0$ and $\vec r .\hat k>0$

$⇒x>0, y > 0$ and $z>0$

$⇒x, y, z$ are positive integers.

$⇒\vec r.\vec a ≤12$

$⇒x + y + z≤12$.

Let $x+y+z+t=12$, where $t≥ 0$.

Clearly, total number of vectors satisfying the given conditions is same as the total number of integral solutions of

$x + y + z+t=12$, where $x > 0, y > 0, z> 0$ and $t≥0$.

Let $x'=x-1, y' =y-1$ and $z' =z-1$. Then, we have

$x'y' +z'+t=9$, where $x' ≥0, y' ≥0,z' ≥0, t≥0$

The total number of solutions of this equation is

${^{9+4-1}C}_{4 -1} ={^{12}C}_3$

Hence, required number of vectors = ${^{12}C}_3$