Question

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

• A. $9$
• B. $12$
• C. $11$
• D. $10$

Answer 1

The correct option is A $9$

Given ${}^n{C}_r=84,{}^n{C}_{r-1}=36$ and ${}^n{C}_{r+1}=126$
Now,
$\frac{{}^n{C}_r}{{}^n{C}_{r-1}}=\frac{84}{36}\Rightarrow \frac{n-r+1}{r}=\frac{7}{3}\cdots \left(1\right)$
Now,
$\frac{{}^n{C}_{r+1}}{{}^n{C}_r}=\frac{126}{84}\Rightarrow \frac{n-r}{r+1}=\frac{3}{2}\Rightarrow n-r=\frac{3(r+1)}{2}\cdots \left(2\right)$
Using equation $\left(1\right)$, we get
$\Rightarrow \frac{3(r+1)}{2}+1=\frac{7r}{3}\Rightarrow 9r+9+6=14r\Rightarrow r=3$
Using equation $\left(2\right)$, we get
$n-3=6\Rightarrow n=9$