Question

If ${}^n{P}_4=24·{}^n{C}_5$then the value of $n$ is

• A. $10$
• B. $15$
• C. $9$
• D. $5$

Answer 1

The correct option is C

$9$

Explanation for correct answer:

Given, ${}^n{P}_4=24.{}^n{C}_5$

$\therefore \frac{n!}{(n-4)!}=24·\frac{n!}{[5!(n-5)!]}$ [${}^{\mathbf{n}}{\mathbf{P}}_{\mathbf{r}\mathbf{}\mathbf{}}\mathbf{=}\frac{\mathbf{n}\mathbf{!}}{\mathbf{(}\mathbf{n}\mathbf{-}\mathbf{r}\mathbf{)}\mathbf{!}}\mathbf{}\mathbf{}$ and ${}^{\mathbf{n}}{\mathbf{C}}_{\mathbf{r}}\mathbf{=}\frac{\mathbf{n}\mathbf{!}}{\mathbf{r}\mathbf{!}\mathbf{(}\mathbf{n}\mathbf{-}\mathbf{r}\mathbf{)}\mathbf{!}}$]

$\Rightarrow \left(n-3\right)\left(n-2\right)\left(n-1\right)n=24·\frac{1}{5!}\left(n-4\right)\left(n-3\right)\left(n-2\right)\left(n-1\right)n$

$\Rightarrow \frac{1}{(n-4)}=\frac{1}{5}$

$\Rightarrow n-4=5$

$\Rightarrow n=9$

Hence, option C is correct.