Let $\overrightarrow{a}=\widehat{i}+\widehat{j}+\widehat{k}$ and let $\overrightarrow{r}$ be a variable vector such  

that $\overrightarrow{r}\cdot \widehat{i},\overrightarrow{r}\cdot \widehat{j}$ and $\overrightarrow{r}\cdot \widehat{k}$ are positive integers. If $\overrightarrow{r}\cdot \overrightarrow{a}\le 12,$ then

the total number of such vectors is

Let $\overrightarrow{r}=x\widehat{i}+y\widehat{j}+z\widehat{k}$ where $x, y, z$ are integers.

We have,

$\overrightarrow{r}\cdot \widehat{i}>0,\overrightarrow{r}\cdot \widehat{j}>0$ and $\overrightarrow{r}\cdot \widehat{k}>0$

$x>0,y>0$ and $z>0$

x, y, z are positive integers.  

Also,

$\begin{array}{r}\overrightarrow{r}\cdot \overrightarrow{a}\le 12\\ \Rightarrow x+y+z\le 12\end{array}$

Let $x+y+z+t=12$ where $t\ge 0$

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 $t\ge 0$. 

Let $x^{\mathrm{\prime }}=x-1,y^{\mathrm{\prime }}=y-1$ and $z^{\mathrm{\prime }}=z-1$ Then we have

$x^{\mathrm{\prime }}+y^{\mathrm{\prime }}+z^{\mathrm{\prime }}+t=9,\text{ where }{x}^{\mathrm{\prime }}\ge 0,y^{\mathrm{\prime }}\ge 0z^{\mathrm{\prime }}\ge 0,t\ge 0$

The total number of solutions of this equation is 

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

Hence, required number of vectors ${=}^{12}{C}_3=220$