Question

${}^n{P}_3:{}^{n-1}{P}_3=5:4$

Answer 1

${}^n{P}_3:{}^{n-1}{P}_3=5:4$
$\begin{array}{c}{}^n{P}_3{:}^{n-1}{P}_3=5:4\\ \frac{{}^n{P}_3}{{}^{n-1}{P}_3}=\frac{5}{4}\\ 4^n{P}_3=5^n{P}_3\\ 4\frac{n!}{\left(n-3\right)!}=5\frac{\left(n-1\right)!}{\left(n-4\right)!}\\ 4\frac{n!}{\left(n-1\right)!}=5\frac{\left(n-3\right)!}{\left(n-4\right)!}\\ 4\frac{n\left(n-1\right)!}{\left(n-1\right)!}=5\frac{\left(n-3\right)\left(n-4\right)!}{\left(n-4\right)!}\\ 4n=5n-15\\ n=15\end{array}$