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Question: Set of k for which 5k - \(1P_{2} + 4(k -^{1}C_{3} - k -^{1}C_{4}) > 0\) is...

Set of k for which 5k - 1P2+4(k−1C3−k−1C4)>01P_{2} + 4(k -^{1}C_{3} - k -^{1}C_{4}) > 0 is

A

{2, 3, 4, 5}

B

{3, 4, 5,6}

C

{4, 5, 6, 7}

D

{5, 6, 7, 8}

Answer

{5, 6, 7, 8}

Explanation

Solution

5(k – 2) (k – 1) + 4

{(k−1)(k−2)(k−3)3x2 6mu− 6mu(k−1)(k−2)(k−3)(k−4)4x3x2}>0\left\{ \frac{(k - 1)(k - 2)(k - 3)}{3x2}\mspace{6mu} - \mspace{6mu}\frac{(k - 1)(k - 2)(k - 3)(k - 4)}{4x3x2} \right\} > 0⇒ {5 6mu+ 6mu4(k−3)6 6mu− 6mu(k−3)(k−4)6}>0\left\{ 5\mspace{6mu} + \mspace{6mu}\frac{4(k - 3)}{6}\mspace{6mu} - \mspace{6mu}\frac{(k - 3)(k - 4)}{6} \right\} > 0

⇒ k2 – 11k – 6 < 0

⇒ 11−1462 6mu<k<11+1462\frac{11 - \sqrt{146}}{2}\mspace{6mu} < k < \frac{11 + \sqrt{146}}{2}

Also to make given terms define k – 1 ≥ 4

K ≥ 5

∴ k = {5, 6, 7, 8, 9, 10, 11}, only (4) contains all these values.