Question

The position vector of a point P is vector r = xi + yj + zk where x, y, z ∈ N and vector a = i + j + k. if vector r.a = 10,  then the number of possible positions of P is 

(A) 30 

(B) 72

(C) 66 

(D) ⁹C₂

Answer 1

Answer is (D) ⁹C₂

Given

vector r.a = 10 ⇒ x + y + z = 10, x, y, x ≥ 1

The number of possible positions of P is

coefficient of x¹⁰ in (x + x² + x³ + ....)³

= coefficient of x⁷ in(1 - x)^-3

= ^{3 + 7 - 1}C₇ = ⁹C₇ = ⁹C₂ = 36