If P(9, r) = 3024, find r. - Vidyadip

Question

If P(9, r) = 3024, find r.

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Answer

We have,
P(9, r) = 3024
${P}(9,\text{r}) = \frac{9!}{(9-\text{r})!}\Big[\because^\text{n}\text{P}_\text{r}=\frac{\text{n!}}{(\text{n}-\text{r})!}\Big]$
$ \Rightarrow\frac{1}{(9-\text{r})!} =\frac{3024}{9\times 8\times7\times6\times5\times4\times3\times2\times1}$
$ \Rightarrow\frac{1}{(9-\text{r})!} =\frac{336}{8\times7\times6\times5\times4\times3\times2\times1}$
$ \Rightarrow\frac{1}{(9-\text{r})!} =\frac{42}{7\times6\times5\times4\times3\times2\times1}$
$ \Rightarrow\frac{1}{(9-\text{r})!} =\frac{42}{7\times6\times5\times4\times3\times2\times1}$
$ \Rightarrow\frac{1}{(9-\text{r})!} =\frac{1}{5\times4\times3\times2\times1}$
$ \Rightarrow\frac{1}{(9-\text{r})!} =\frac{1}{5!}$
$\Rightarrow (9-\text{r})! =5!$
$\Rightarrow 9-\text{r}= 5$
$\Rightarrow 9-5=\text{r}$
$\Rightarrow 4=\text{r}$
$\Rightarrow \text{r}= 4$
Hence, $ \text{r}= 4 $

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