Q39 / 110 · VITEEE 2024 · Mathematics
If the coefficients of $x^7$ and $x^8$ in the expansion of $\left[2+\frac{x}{3}\right]^n$ are equal, then value of $n$ is
- A15
- B45
- Correct answerC55
- D56
Solution
Coefficient of $x^7={ }^n C_7(2)^{n-7}\left(\frac{1}{3}\right)^7$ and Coefficient of $x^8={ }^n C_8 \cdot(2)^{n-8}\left(\frac{1}{3}\right)^8$
According to the question,
$\begin{aligned}
& { }^n C_7(2)^{n-7}\left(\frac{1}{3}\right)^7={ }^n C_8(2)^{n-8}\left(\frac{1}{3}\right)^8 \\
\Rightarrow & \frac{n!}{7!(n-7)!} \cdot \frac{2^{n-7}}{2^{n-8}}=\frac{1}{3} \cdot \frac{n!}{8!(n-8)!} \\
\Rightarrow & 2=\frac{7!}{8!} \cdot \frac{(n-7)!}{(n-8)!} \cdot \frac{1}{3} \\
\Rightarrow & 2=\frac{1}{8} \times(n-7) \times \frac{1}{3} \Rightarrow n-7=48 \\
\Rightarrow & n=55
\end{aligned}$
Hence, the correct answer is option: 'C'.