Question

If $P(5,r)=2.P(6,r-1)$, find $r$.

Answer 1

$P(5,r)=2.P(6,r-1)$

$\Rightarrow \frac{5!}{(5-r)!}=2.\frac{6!}{(6-(r-1\left)\right)!}$

$\Rightarrow \frac{5!}{(5-r)!}=\frac{2\times 6\times 5!}{(7-r)!}$

$\Rightarrow \frac{5!}{(5-r)!}=\frac{12\times 5!}{(7-r)(6-r)(5-r)!}$

$\Rightarrow 1=\frac{12}{(7-r)(6-r)}$

$\Rightarrow (7-r)(6-r)=4\times 3$

$\Rightarrow r=3$