Question

If ²ⁿC₃ : ⁿC₂ = 12 : 1, then the value of n is ?

1. 6
2. 5
3. 4
4. 3

Answer 1

Correct Answer - Option 2 : 5

Given:

If 2nC3 : nC2 = 12 : 1

Formula used :

ⁿC_r = n!/r!(n - r)!

Calculation:

If 2nC3 : nC2 = 12 : 1      ----(1)

Using the formula

²ⁿC₂ = {2n!/3! (2n - 3)!}

⇒ 2n(2n - 1)(2n - 2)(2n - 3)!/(2n - 3)! × 3 × 2

⇒ n(2n - 1)2(n - 1)/3      ----(2)

ⁿC₂ = {n!/2! (n - 2)!

⇒ n(n - 1)(n - 2)!/(n - 2)! × 2

⇒ n(n - 1)/2      ----(3)

Putting equation 2 and 3 in equation 1

⇒ {n(2n - 1)2(n - 1)/3}/{n(n - 1)/2} = 12/1

⇒ 2n - 1 = 9

⇒ 2n = 10 

⇒ n = 5

∴ The value of n is 5.