Question

Find the value of $n$ by solving $\frac{{}^n{P}_4}{{}^{n-1}{P}_4}=\frac{5}{3}$ where, $n>4$.

• A. $5$
• B. $10$
• C. $15$
• D. $20$

Answer 1

The correct option is B $10$

$\frac{{}^n{P}_4}{{}^{n-1}{P}_4}=\frac{5}{3},n>4$

$\Rightarrow \frac{\frac{n!}{(n-4)!}}{\frac{(n-1)!}{(n-1-4)!}}=\frac{5}{3}$

$\Rightarrow \frac{\frac{n!}{(n-4)!}}{\frac{(n-1)!}{(n-5)!}}=\frac{5}{3}$

$\Rightarrow \frac{n!}{(n-1)!}\times \frac{(n-5)!}{(n-4)!}=\frac{5}{3}$

$\Rightarrow \frac{n(n-1)!}{(n-1)!}\times \frac{(n-5)!}{(n-4)(n-5)!}=\frac{5}{3}$

$\Rightarrow \frac{n}{n-4}=\frac{5}{3}$

$\Rightarrow 3n=5n-20$

$\Rightarrow n=10$