Question

Find the value of $n$ and $r$, if ${}^n{P}_r=720$ and ${}^n{C}_r=120$.

Answer 1

$n_{P_r}=r!n_{C_r}$

$\Rightarrow \frac{n_{P_r}}{n_{C_r}}=r!\Rightarrow \frac{720}{120}=r!$

$\Rightarrow r=3$

Now, $n_{P_3}=720$

$\Rightarrow n(n-1)(n-2)=720=10\times 9\times 8$

$\Rightarrow n=10$