Question

If the coefficients of $x^7$ and $x^8$ in ${(2+\frac{x}{3})}^n$ are equal then n $=$

• A. 45
• B. 55
• C. 35
• D. 27

Answer 1

The correct option is B 55

Coefficient of $x^7 = coefficient of $x^8

$\Rightarrow \frac{2^n{\cdot }^n{C}_7}{6^7}=2^n\cdot \frac{{}^n{C}_8}{6^8}$
$\Rightarrow 6{(}^n{C}_7){=}^n{C}_8$
$\Rightarrow 6\cdot \frac{n!}{(n-7)!7!}=\frac{n!}{(n-8!)8!}$
$\Rightarrow \frac{6}{n-7}=\frac{1}{8}\Rightarrow n-7=48$
$\Rightarrow n=55$