Question

The number of positive integers satisfying the inequality `""^(n+1)C_(n-2) - ""^(n+1)C_(n-1) ≤ 100` is ______.

Options

• Nine

• Eight

• Five

• Ten

Answer 1

The number of positive integers satisfying the inequality `""^(n+1)C_(n-2) - ""^(n+1)C_(n-1) ≤ 100` is nine.

Explanation:

`""^((n+1))C_((n-2)) - ""^((n+1))C_((n-1)) ≤ 100`

⇒ `((n + 1)!)/((n - 2)!(n + 1 - n + 2)!) - ((n + 1)!)/((n - 1)!(n + 1 - n + 1)!) ≤ 100`

⇒ `((n + 1)!)/((n - 2)!3!) - ((n + 1)!)/((n - 1)!2!) ≤ 100`

⇒ `(n + 1)![1/(3!(n - 2)!) - 1/(2!(n - 1)!)] ≤ 100`

⇒ `(n + 1)![((n - 1)! - 3(n - 2)!)/(6(n - 1)!(n - 2)!)] ≤ 100`

⇒ `((n + 1)[(n - 1)(n - 2)! -3(n - 2)!])/(6(n - 1)!(n - 2)!) ≤ 100`

⇒ `(n(n + 1)[(n - 1)(n - 2)!(n - 1 - 3)])/(6(n - 1)!(n - 2)!) ≤ 100`

⇒ `(n(n + 1)[(n - 1)(n - 2)!(n - 4)])/(6(n - 1)!(n - 2)!) ≤ 100`

⇒ n(n + 1)(n – 4) ≤ 600

Put n = 1, 2, 3, 4, 5, 6, 7, 8, 9, ......

For n = 5 ⇒ 600 ≥ 5 × 6 × 1 Holds good

For n = 8 ⇒ 600 ≥ 8 × 9 × 4 = 288 Holds good

For n = 9 ⇒ 600 ≥ 9 × 10 × 5 = 450 Holds good

For n = 10 ⇒ 600 ≤ 10 × 11 × 7 = 770 Does not hold

Hence, the number of positive integers which is satisfying the given in quality must be 9.