Find n if

Question:

Find $n$ if ${ }^{n-1} \mathrm{P}_{3}:{ }^{\pi} \mathrm{P}_{4}=1: 9$

Solution:

${ }^{n-1} \mathrm{P}_{3}:{ }^{n} \mathrm{P}_{4}=1: 9$

$\Rightarrow \frac{{ }^{n-1} P_{3}}{{ }^{n} P_{4}}=\frac{1}{9}$

$\Rightarrow \frac{\left[\frac{(\mathrm{n}-1) !}{(\mathrm{n}-1-3) !}\right]}{\left[\frac{\mathrm{n} !}{(\mathrm{n}-4) !}\right]}=\frac{1}{9}$

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

$\Rightarrow \frac{(n-1) !}{n \times(n-1) !}=\frac{1}{9}$

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

$\therefore \mathrm{n}=9$

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