Question

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

• A. $56$
• B. $55$
• C. $45$
• D. $15$

Answer 1

Answer: (B)

$T_8={}^n{C}_7{\left(2\right)}^{n-7}{\left(\frac{x}{3}\right)}^7$
$={}^n{C}_7\frac{2^{n-7}}{3^7}{x}^7$
and $T_9={}^n{C}_8{\left(2\right)}^{n-8}{\left(\frac{x}{3}\right)}^8$
$={}^n{C}_8\frac{2^{n-8}}{3^8}{x}^8$
Now, it is given that ${}^n{C}_7\frac{{\left(2\right)}^{n-7}}{3^7}={}^n{C}_8\frac{2^{n-8}}{3^8}$
$\Rightarrow \frac{8}{n-7}=\frac{1}{6}$
$\Rightarrow n=55$