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Permutation ratio identity: If (2n + 1)P(n − 1) : (2n − 1)P(n) = 3 : 5, find n.

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.

Permutation ratio identity: If (2n + 1)P(n − 1) : (2n − 1)P(n) = 3 : 5, find n.

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