MathematicsIf (n+1)P5:nP6=2:7, then the value of n is

If $^{(n + 1)} P_{5} :^{n} P_{6} = 2 : 7,$ then the value of n is

A
13
B
11
C
15
D
17

Solution:

\frac{^{n + 1} P_{5}}{^{n} P_{6}} = \frac{2}{7} \Rightarrow 7 (n + 1) = 2 (n^{2} - 9 n + 20) \Rightarrow 7 n + 7 = 2 n^{2} - 18 n + 40

$\Rightarrow 2 n^{2} - 25 n + 33 = 0$

$\Rightarrow 2 n^{2} - 22 n - 3 n + 33 = 0 \Rightarrow 2 n (n - 11) - 3 (n - 11) = 0$

$\Rightarrow (n - 11) (2 n - 3) = 0 \Rightarrow n = 11, \frac{3}{2}$

but n is a positive integer $\Rightarrow n = 11.$

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