Question:
If ${ }^{n} P_{r}={ }^{n} P_{r+1}$ and ${ }^{n} C_{r}={ }^{n} C_{r-1}$, then the value of $r$ is equal to:
Correct Option: , 3
Solution:
${ }^{\mathrm{n}} \mathrm{P}_{\mathrm{r}}={ }^{\mathrm{n}} \mathrm{P}_{\mathrm{r}+1} \Rightarrow \frac{\mathrm{n} !}{(\mathrm{n}-\mathrm{r}) !}=\frac{\mathrm{n} !}{(\mathrm{n}-\mathrm{r}-1) !}$
$\Rightarrow(\mathrm{n}-\mathrm{r})=1$ ..........(1)
${ }^{n} \mathrm{C}_{r}={ }^{n} \mathrm{C}_{r-1}$
$\Rightarrow \frac{n !}{r !(n-r) !}=\frac{n !}{(r-1) !(n-r+1) !}$
$\Rightarrow \frac{1}{r(n-r) !}=\frac{1}{(n-r+1)(n-r) !}$
$\Rightarrow \mathrm{n}-\mathrm{r}+1=\mathrm{r}$
$\Rightarrow \mathrm{n}+1=2 \mathrm{r}$ .............(2)
$(1) \Rightarrow 2 r-1-r=1 \Rightarrow r=2$