Question

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

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

Answer 1

The correct option is B $55$

It is given that the coefficients $T_8=T_9$
Therefore ${}^n{C}_7{2}^{n-7}{3}^{-7}={}^n{C}_8{2}^{n-8}{3}^{-8}$
$6{}^n{C}_7={}^n{C}_8$
$6\left[\frac{n!}{(n-7)!7!}\right]=\frac{n!}{(n-8)!8!}$
$\frac{6}{n-7}=\frac{1}{8}$
$n-7=48$
$\therefore n=55$