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

Answer: (B)

We have, $(1+x{)}^n$
$T_9={}^n{C}_8{x}^8$
$T_7={}^n{C}_6{x}^6$
Given, ${}^n{C}_8:{}^n{C}_6=9:7$
$\Rightarrow \frac{{}^n{C}_8}{{}^n{C}_6}=\frac{9}{7}$
$\Rightarrow \frac{n!6!\left(n-6\right)!}{8!\left(n-8\right)!n!}=\frac{9}{7}$
$\frac{\left(n-6\right)\left(n-7\right)}{8\times 7}=\frac{9}{7}$
$\Rightarrow \left(n-6\right)\left(n-7\right)=9\times 8$
$n=15$
Coefficient of $4$th term of ${\left(1+x\right)}^{15}$
${}^{15}{C}_3=\frac{15!}{3!12!}=455$