Question

Find $r$ if
${}^5{P}_r=2{}^6{P}_{r-1}$

Answer 1

Consider the given equation,

${}^5{P}_r=2\left({}^6{P}_{r-1}\right)$

$\frac{5!}{(5-r)!r!}=\frac{2.6!}{(6-r+1)!(r-1)!}$

$\frac{5!}{(5-r)!r(r-1)!}=\frac{2.6!}{(5-r)!(r-1)!}$

$\frac{5!}{r}=\frac{2.6!}{1}$

$r=\frac{5!}{6!}=\frac{2}{6}=\frac{1}{3}$

Hence, this is the answer.