Question:

If \(P(2n+1,n-1):P(2n-1,n) = 3:5\), find \(n\).

Show Hint

Simplify factorials carefully; permutation ratios often reduce to quadratic equations.
Updated On: Sep 30, 2025
  • 2
  • 4
  • 6
  • 8
  • 10
Hide Solution
collegedunia
Verified By Collegedunia

The Correct Option is B

Solution and Explanation

Step 1: Formula.
\[ P(n,r) = \frac{n!}{(n-r)!} \]
Step 2: Expand ratio.
\[ \frac{P(2n+1,n-1)}{P(2n-1,n)} = \frac{3}{5} \] \[ \frac{(2n+1)!}{(n+2)!} \cdot \frac{(n-1)!}{(2n-1)!} = \frac{3}{5} \]
Step 3: Simplify.
\[ \frac{(2n+1)(2n)}{(n+2)(n+1)n} = \frac{3}{5} \]
Step 4: Cross-multiply.
\[ 5(2n+1)(2n) = 3(n+2)(n+1)n \] \[ 20n^2 + 10n = 3n^3 + 9n^2 + 6n \] \[ 3n^3 - 11n^2 - 4n = 0 \]
Step 5: Solve quadratic.
\[ n(3n^2 - 11n - 4) = 0 \] \[ n = 4 \quad (\text{valid}) \]
Final Answer:
\[ \boxed{4} \]
Was this answer helpful?
0
0