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The position vector of a point P is $\overset{⇀}{r} = x \overset{⇀}{i} + y \overset{⇀}{j} + z \overset{⇀}{k}$ where x and y are positive integers and $\overset{⇀}{a} = \overset{⇀}{i} + \overset{⇀}{j} + \overset{⇀}{k}$ . If $\overset{⇀}{r} . \overset{⇀}{a} = 10$ , then the number of possible positions of P is :

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$48$

$72$

$24$

$36$

answer is D.

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Detailed Solution

$\vec{r} . \vec{a} = x + y + z$

$∴$ x + y + z = 10 where x and y are positive integers.
$∴$ No. of positive integral solutions of
$x + y + z = 10 i s^{10 - 1} C_{3 - 1} =^{9} C_{2} = 36$ .

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