Expansion of Binomial Coefficients

Given that the coefficient of $x^3$ is 3 times that of $x^2$ in the expansion $(2 + 3x)^n$, find the value of $n$.

This problem focuses on binomial expansions and the use of binomial coefficients. When exploring binomial expansions, it's crucial to understand how the coefficients are derived and how the powers of different terms come together in the expression. In the expansion of a binomial expression like $(2 + 3x)^n$, each term can be expressed using the binomial theorem, which relies on binomial coefficients. The binomial coefficient, denoted as $C(n, k)$, is used to find the coefficients of $x$ in each term of the expansion.

In this problem, you're asked to compare coefficients of different powers of $x$, specifically $x^3$ and $x^2$. Understanding how to work with coefficients involves recognizing patterns and knowing how they relate to each other in various powers. A key strategy here is using relationships between coefficients to set up a system of equations that can be solved for $n$, effectively applying algebraic manipulation and reasoning. Combinatorial reasoning can sometimes come into play, especially when identifying how different terms, once expanded, contribute to the coefficient of a specific power. Exploring this problem offers insight into how binomial expansions are not only about algebraic techniques but also numerical reasoning and combinatorial logic.