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Permutations — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsPermutations3 Marks
Question
If P(n − 1, 3) : P(n, 4) = 1 : 9, find n.
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Answer

We have,
P(n − 1, 3) : P(n, 4) = 1 : 9
$\Rightarrow \frac{\text{p}(\text{n-1,3})}{\text{P} (\text{n}, 4)}=\frac{1}{9}$
$\Rightarrow\frac{\frac{\text{(n-1)!}}{(\text{(n-1-3)!})}}{\frac{\text{n}!}{\text{(n-4)!}}}=\frac{1}{9}$
$\Rightarrow \frac{\text{(n-1) }\times\text{(n-4)!}}{\text{(n-4)!}\times\text{n}!}=\frac{1}{9}$
$\Rightarrow \frac{\text{(n-1)}}{\text{n!}}=\frac{1}{9}$
$\Rightarrow \frac{\text{(n-1)!}}{\text{n}\times(\text{n-1})!}=\frac{1}{9}$
$​​\Rightarrow\frac{1}{\text{n}}=\frac{1}{9}$
$\Rightarrow\text{n}=9$
Hence, $\text{n}=9$

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