If the position vector of a point P is $\to r = x^i = y^j + z^k,$ where x, y, z $ϵ$ N and projection of $\to r$ on $\to a =^i +^j +^k i s \frac{10}{\sqrt{3}}$ then number of possible of P is also equal to

number of natural solution of the equation x + y + z = 7.

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number of ways of selecting two objects out of 10 distinct objects arranged in a row so that no two of them are next to each other

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the total number of outcomes when a pair of dice are rolled once.

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number of positive divisors of 1800.

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Solution

The correct options are
B
number of ways of selecting two objects out of 10 distinct objects arranged in a row so that no two of them are next to each other

C
the total number of outcomes when a pair of dice are rolled once.

D
number of positive divisors of 1800.

Projection of $\to r o n \to a i s \frac{\to r . \to a}{| \to a |} \Rightarrow x + y + z = 10 (x, y, z ϵ N)$ number of solution $=^{10 - 1} C_{3 - 1} =^{9} C_{2} = 36$

(a) x + y + z = 7, x, y, z $\geq$ 1

$\Rightarrow X + Y + Z = 4; where X = x + 1, Y = y + 1, Z = z + 1 \Rightarrow number of solution =^{6} C_{2} = 15$

(b) selecting any 2 out of 10 = $^{10} C_{2}$ ways

selecting 2 next to each other = 9 ways

Therefore, number of ways of selecting two objects out of 10 distinct objects arranged in a row so that no two of them are next to each other = $^{10} C_{2} - 9 = 36$

$(c) 6^{2} = 36$

$(d) 1800 = 2^{3} 3^{2} 5^{2} ∴ no. of divisors = (3 + 1) (2 + 1) (2 + 1) = 36$

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