First slide

Permutations

Question

If $^{2 n + 1} P_{n - 1} :^{2 n - 1} P_{n} = 3 : 5$ then the value of n is equal to

Moderate

A

3
B

4
C

5
D

6

Solution

We have, $^{2 n + 1} P_{n - 1} :^{2 n - 1} P_{n} = 3 : 5$
$\begin{array}{l} \Rightarrow \frac{^{2 n + 1} P_{n - 1}}{2 n - 1_{P_{n}}} = \frac{3}{5} \\ \Rightarrow \frac{(2 n + 1)!}{(n + 2)!} \times \frac{(n - 1)!}{(2 n - 1)!} = \frac{3}{5} \\ \Rightarrow \frac{(2 n + 1) (2 n)}{(n + 2) (n + 1) n} = \frac{3}{5} \\ \Rightarrow 10 (2 n + 1) = 3 (n + 2) (n + 1) \\ \Rightarrow 3 n^{2} - 11 n - 4 = 0 \\ \Rightarrow (n - 4) (3 n + 1) = 0 \Rightarrow n = 4 \end{array}$

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