Find r if (i) ⁵Pᵣ= 2 ⁶Pᵣ-₁(ii) ⁵Pᵣ= ⁶Pᵣ-₁

Solution:

(i) Given that, ⁵ Pᵣ= 2 ⁶Pᵣ-₁

Using nPr formula,

5!/(5 - r)! = 2 × 6!/[6 - (r - 1)]!

5!/(5 - r)! = 2 × 6 × 5!/(7 - r)!

5!/(5 - r)! = 2 × 6 × 5!/[(7 - r)(6 - r)(5 - r)!]

(7 - r)(6 - r) = 12

42 - 7r - 6r + r² = 12 

r² - 13r + 30 = 0

(r - 3) (r - 10) = 0

r = 3, 10.

Since we are given with ⁵ Pᵣ and ⁶Pᵣ-₁ in the problem, r ≤ 5 and r-1 ≤ 6. Thus, r cannot be 10.

Thus, r = 3.

(ii) Given that, ⁵ Pᵣ = ⁶ Pᵣ-₁ .

Using nPr formula,

5!/(5 - r)! = 6!/[6 - (r - 1)]!

5!/(5 - r)! = 6 × 5!/(7 - r)!

5!/(5 - r)! = 6 × 5!/[(7 - r)(6 - r)(5 - r)!]

(7 - r)(6 - r) = 6

42 - 7r - 6r + r² = 6 

r² - 13r + 36 = 0

(r - 4) (r - 9) = 0

r = 4, 9.

Since we are given with ⁵ Pᵣ and ⁶Pᵣ-₁ in the problem, r ≤ 5 and r-1 ≤ 6. Thus, r cannot be 9.

Thus, r = 4

---

NCERT Solutions Class 11 Maths Chapter 7 Exercise 7.3 Question 7

Find r if (i) ⁵Pᵣ= 2 ⁶Pᵣ-₁(ii) ⁵Pᵣ= ⁶Pᵣ-₁

Summary:

(i) If ⁵Pᵣ= 2 ⁶Pᵣ-₁, then r = 3 (ii) If ⁵Pᵣ= ⁶Pᵣ-₁, then r = 4