Question

If $^{n + 2}C_{8}:^{n - 2}P_{4}=57:16$ , then the value of $n$ is

Answer 1

$\frac{\left(n + 2\right) !}{8 ! \left(n - 6\right) !}\times \frac{\left(n - 6\right) !}{\left(n - 2\right) !}=\frac{57}{16}$
$\Rightarrow \left(n + 2\right)\left(n + 1\right)n\left(n - 1\right)=\frac{57}{16}\times 8!$
$\Rightarrow \left(n - 1\right)n\left(n + 1\right)\left(n + 2\right)=57\times 7\times 6\times 5\times 4\times 3$
$\Rightarrow \left(n - 1\right)n\left(n + 1\right)\left(n + 2\right)=18\times 19\times 20\times 21$
$\Rightarrow n=19$