Find n if $^{n - 1} P_{3} :^{n} P_{4} = 1 : 9$ .

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Find n if

^{n - 1} P_{3} :^{n} P_{4} = 1 : 9

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Text solutionVerified

Given,

^{n - 1} P_{3} :^{n} P_{4} = 1 : 9

\Rightarrow \frac{^{n - 1} P _{3}}{^{n} P _{4}} = \frac{1}{9}

\Rightarrow \frac{\frac{( n - 1 )!}{( n - 1 - 3 )!}}{\frac{n !}{( n - 4 )!}} = \frac{1}{9}

\Rightarrow \frac{( n - 1 )!}{( n - 4 )!} \cdot \frac{( n - 4 )!}{n !} = \frac{1}{9}

\Rightarrow \frac{1}{n} = \frac{1}{9}

\Rightarrow n = 9

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