Mathematics

Problem

Let

\vec{a} = \hat{i} + \hat{j} + \hat{k}

and

\vec{r}

be a variable vector such that

\vec{r} \cdot \hat{i}

\vec{r} \cdot \hat{j}

and

\vec{r} \cdot \hat{k}

are positive integers. If

\vec{r} \cdot \vec{a} \le 12

, then the number of values of

\vec{r}

is (a)

{}^{11}C_2

(b)

{}^{14}C_2

(c)

{}^{15}C_3

(d)

{}^{12}C_3