Question

Find $r$ if
${}^5{P}_r={}^6{P}_{r-1}$

Answer 1

Consider the given equation.

${}^5{P}_r{=}^6{P}_{r-1}$

$\frac{5!}{(5-r)!r!}=\frac{6!}{(6-r+1)!(r-1)!}$

$\frac{5!}{(5-r)!r(r-1)!}=\frac{6!}{(5-r)!(r-1)!}$

$\frac{5!}{r}=\frac{6!}{1}$

$r=\frac{5!}{6!}=\frac{1}{6}$

Hence, this is the answer.