Question

The ratio of coefficients of $9^{th}$ and $7^{th}$ terms in the expansion of $(1+x{)}^n$ is $9:7.$Then the coefficient of $4^{th}$ term is

• A. $395$
• B. $455$
• C. $530$
• D. $645$

Answer 1

The correct option is B $455$

$\frac{\text{Coefficient of}{T}_9}{\text{Coefficient of}{T}_7}=\frac{{}^n{C}_8}{{}^n{C}_6}=\frac{\frac{n!}{8!(n-8)!}}{\frac{n!}{6!(n-6)!}}=\frac{9}{7}$
$\Rightarrow \frac{(n-6)(n-7)}{8\cdot 7}=\frac{9}{7}\Rightarrow n^2-13x+42=72\Rightarrow n=15,-2\text{(not possible)}$
Hence coefficient of $4^{th}$ term $={}^{15}{C}_3=455$