Difficulty: Medium
Correct Answer: 4
Explanation:
Introduction / Context:We compare two permutations via factorial forms and simplify to a rational function of n. Solving the resulting polynomial yields n.
Given Data / Assumptions:(2n + 1)P(n − 1) : (2n − 1)P(n) = 3 : 5.
Concept / Approach:Use P(a, b) = a! / (a − b)!. Compute the ratio explicitly and reduce. Then equate to 3/5 and solve for n in integers.
Step-by-Step Solution:
R = [(2n + 1)! / (n + 2)!] / [(2n − 1)! / (n − 1)!] = [(2n)(2n + 1)] / [(n + 2)(n + 1)n]Set R = 3/5 ⇒ 5(2n)(2n + 1) = 3n(n + 1)(n + 2)Expand ⇒ 20n^2 + 10n = 3n^3 + 9n^2 + 6n ⇒ 3n^3 − 11n^2 − 4n = 0Factor ⇒ n(3n^2 − 11n − 4) = 0 ⇒ n = 4 (positive integer root)Verification / Alternative check:Substitute n = 4 into the original ratio and both sides simplify to 3/5.
Why Other Options Are Wrong:6, 3, or 8 do not satisfy the polynomial; negative roots are not admissible.
Common Pitfalls:Dropping factorial cancellations incorrectly, or mishandling the polynomial algebra.
Final Answer:4
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