Question

If the coefficient of $x^7$ & $x^8$ in the expansion of ${[2+\frac{x}{3}]}^n$ are equal, then the value of $n$ is :

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

Answer 1

Answer: (C)

$55$

coefficient of $x^7$ in $[2+\frac{x}{3}{]}^n$ is ${}^n{C}_7\left(2{)}^{n-7}\right(\frac{1}{3}{)}^7$

coefficient of $x^8$ in $[2+\frac{x}{3}{]}^n$ is ${}^n{C}_8\left(2{)}^{n-8}\right(\frac{1}{3}{)}^8$

$⟹{}^n{C}_7\times \left(2^{n-7}\right)\times \frac{1}{3^7}{=}^n{C}_8\times \left(2^{n-8}\right)\times \frac{1}{3^8}$

$⟹\frac{{}^n{C}_8}{{}^n{C}_7}=6$

$⟹\frac{n-7}{8}=6$

$⟹n=55$