If p , q , r ∈ N , then the number of points having position vector p i ^ + q j ^ + r k ^ such that 8 ≤ p + q + r ≤ 12 are

Step-by-step solution

8 ≤ p + q + r ≤ 1 2 ⇒ p + q + r = 8,9 , 10,11,1 2 Here, p , q , r ≥ 1 Hence, required number using number of positive integral solutions = n - 1 C r - 1 = 7 C 2 + 8 C 2 + 9 C 2 + 10 C 2 + 11 C 2 = 7 × 6 2 + 8 × 7 2 + 9 × 8 2 + 10 × 9 2 + 11 × 10 2 = 21 + 28 + 36 + 45 + 5 5 = 18 5