Question

If $462,330$ and $165$ are three successive coefficients in the expansion of $(1+x{)}^n$, then $n=$

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

Answer 1

The correct option is C $11$

It is given that $T_r=462$ $T_{r+1}=330$ and $T_{r+2}=165$

Therefore $\frac{T_{r+1}}{T_{r+2}}=2$

$T_{r+1}=2T_{r+2}$

$\frac{r+1}{n-r}=2$

$r+1=2n-2r$

$3r=2n-1$...(i)

$\frac{T_r}{T_{r+1}}=\frac{7}{5}$

$\frac{r}{n+1-r}=\frac{7}{5}$

$5r=7n+7-7r$

$12r=7n+7$ ...(ii)

$Eq\left(ii\right)-4Eq\left(i\right)$ implies

$7n+7=8n-4$

$n=11$