Permutations & Combinations
Permutation ratio equation
nta_pyq_2023_jan
Grade 11
Question:
If \(^{2n+1}P_{n-1} : {}^{2n-1}P_n = 11 : 21\), then \(n^2 + n + 15\) is equal to:
Step-by-Step Solution
Key Concept: Expand the permutation ratio using the formula \(^m P_r = \frac{m!}{(m-r)!}\) and simplify.
\(\frac{(2n+1)!(n+2)!}{(n+2)!(2n-1)!} = \frac{11}{21}\) $\to$ \(\frac{(2n+1)(2n)}{(n+2)(n+1)n} = \frac{11}{21}\) $\to$ \(\frac{2n+1}{(n+1)(n+2)} = \frac{11}{42}\) $\to$ n=5. So n²+n+15 = 25+5+15 = 45.
Correct Answer: 45