Given, ^nP_4^n 1P_5=3 or, n(n 1)(n 2)(n 3)(n 1)(n 2)(n 3)(n 4)(n 5)=3 or, n=3(n 4)(n 5) or, 3n^2 27n+60=n or, 3n^2 28n+60=0 or, 3n^ ...

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Synthetic Division of Polynomials

Solve: nP4n-...

Question

Solve:
$\frac{^{n} P_{4}}{^{n - 1} P_{5}} = 3$ .

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Solution

Given,

$\frac{^{n} P_{4}}{^{n - 1} P_{5}} = 3$

or, $\frac{n (n - 1) (n - 2) (n - 3)}{(n - 1) (n - 2) (n - 3) (n - 4) (n - 5)} = 3$

or, $n = 3 (n - 4) (n - 5)$

or, $3 n^{2} - 27 n + 60 = n$

or, $3 n^{2} - 28 n + 60 = 0$

or, $3 n^{2} - 18 n - 10 n + 60 = 0$

or, $3 n (n - 6) - 10 (n - 6) = 0$

or, $(3 n - 10) (n - 6) = 0$ .

Since $n$ is a natural number then $n = 6$ .

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n

\frac{^{n} P_{4}}{^{n - 1} P_{4}} = \frac{5}{3}

n > 4

^{n} P_{5} :^{n - 1} P_{5} = 3 : 2

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^{n} P_{5} : n - 1 P_{5} = 3 : 2

n

^{n}^{+}^{1} P_{5} :^{n} P_{6} =

2 : 7

n

=

A :^{n} P_{5} =^{n} P_{6} \Rightarrow n =

B :^{n} P_{12} =^{n} P_{8} \Rightarrow n =

C :^{n} C_{(n - 3)} = 10 \Rightarrow n =

D :^{(n + 1)} P_{5} :^{n} P_{6} = 1 : 2 \Rightarrow n =

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