First slide

Permutations

Question

If $2 n + 1_{P_{n - 1}} :^{2 n - 1} P_{n} = 3 : 5$ then equal to

Moderate

A

4
B

6
C

3
D

8

Solution

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

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