If the position vector of a point P is $\vec r=x\hat i+y\hat j+z\hat k$ where $x, y, z ∈N$ and $\vec a$ is a vector given by $\vec a = \hat i +\hat j+\hat k$, then the total number of possible positions of point P for which $\vec r.\vec a = 10$, is

36

The correct answer is Option (1) → 36

$\vec r.\vec a = 10⇒ x + y + z=10$

as $x,y,z∈N$

assuming 1 to each variables

now we need to distribute $(10-3)=7$ among $x,y,z$

Bagger's Method

No. of ways of doing

So using 2 distribution partitions

$=\frac{(7+2)!}{2!7!} =\frac{9!}{2!7!}= 36$.