Question

If ${}^n{P}_r{=}^n{P}_{r+1}$ and ${}^n{C}_r{=}^n{C}_{r-1}$, find n and r.

Answer 1

${}^n{P}_r{=}^n{P}_{r+1}$

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

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

$n-r=1$

$\Rightarrow n=r+1$------------------1

${}^n{C}_r{=}^n{C}_{r-1}$

$\Rightarrow \frac{n!}{r!(n-r)!}=\frac{n!}{(r-1)!(n-r+1)!}$

$\Rightarrow \frac{r!(n-r)!}{(r-1)!(n-r+1)!}=1$

$\Rightarrow \frac{r}{n-r+1}=1$

$r=n-r+1$

$n=2r-1$

From Equation 1, $n=r+1$

$\Rightarrow r+1=2r-1$

$\Rightarrow r=2$

$\Rightarrow n=r+1=3$

$\therefore n=3,r=2$