Question Detail
Question
What is the value of n, if P(15, n − 1) : P(16, n − 2) = 3 : 4?
Correct Answer
Option C is the correct answer.
Detailed Explanation
Given the ratio 15Pn-1 : 16Pn-2 = 3 : 4.This can be written as [15! / (15 - (n-1))!] / [16! / (16 - (n-2))!] = 3/4.Simplify the factorials: [15! / (16-n)!] * [(18-n)! / 16!] = 3/4.(18-n)(17-n)(16-n)! / [(16-n)! * 16] = 3/4.(18-n)(17-n) / 16 = 3/4.Multiply both sides by 16: (18-n)(17-n) = 12.Expand: n2 - 35n + 306 = 12.n2 - 35n + 294 = 0.Factoring the quadratic equation: (n - 14)(n - 21) = 0.This gives n = 14 or n = 21.For permutations kPr, it must be that r ≤ k. So, n-1 ≤ 15 (n ≤ 16) and n-2 ≤ 16 (n ≤ 18).Therefore, n = 14 is the valid solution.
Hint
Recall the definition of permutation nPr = n! / (n-r)! and simplify the ratio of factorials. Pay attention to the conditions for n for permutations.