Step-by-Step Solution
Key Concept: Using the binomial theorem to relate coefficients and solve for the power of 2
The given expression is $\left(x + \frac{3b}{x}\right)C_0 + \left(x + \frac{3b}{x}\right)C_1 + \left(x + \frac{3b}{x}\right)C_2 + \ldots + \left(x + \frac{3b}{x}\right)C_n$. If $P$ is the coefficient of $x^n$, then $P = 2^{3b}C_0 + 2^{3b}C_1 + 2^{3b}C_2 + \ldots + 2^{3b}C_n$. By adding equations and applying the binomial theorem, we get $2P = 2^{3b+1}$, so $P = 2^{3b}$. Therefore, $2^{3b} = 8$.
Correct Answer: 8