Question

If ${}^n{C}_{r-1}=36{,}^n{C}_r=84$ and ${}^n{C}_{r+1}=126$, then $r$ is

• A. $1$
• B. $2$
• C. $3$
• D. $4$

Answer 1

The correct option is C $3$

${}^n{C}_{r-1}=36{,}^n{C}_r=84{,}^n{C}_{r+1}=126$

We know that

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

$\Rightarrow \frac{36}{84}=\frac{r}{n-r+1}$

$\Rightarrow \frac{r}{n-r+1}=\frac{3}{7}$

$\Rightarrow 3n-10r+3=0...\left(1\right)$

Also,

$\frac{{}^n{C}_r}{{}^n{C}_{r+1}}=\frac{r+1}{n-r}=\frac{84}{126}=\frac{2}{3}$

$\Rightarrow 2n-5r-3=0...\left(2\right)$

Solving (1) and (2), we get $n=9$ and $r=3$.