Find Coefficient in Binomial Expansion

Find the coefficient of $x^5$ in the binomial expansion of $(2x - 8)^8$.

The task at hand is to find the coefficient of a specific term in a binomial expansion using the Binomial Theorem. This theorem is a powerful tool in discrete mathematics that allows you to expand expressions of the form $(a + b)^n$ into a sum involving terms of the form $C(n, k) \cdot a^{n-k} \cdot b^k$, where $C(n, k)$ represents the binomial coefficient. In this problem, you are tasked with identifying the coefficient of $x^5$ in the expansion of $(2x - 8)^8$, which involves applying the theorem to determine the appropriate term. Combinatorial reasoning is frequently used in such problems to identify which term's expansion results in a power of $x^5$. This involves balancing the exponents of the constituent parts, in this case, 2x and -8. After determining which term contributes $x^5$, you then compute the corresponding binomial coefficient, which is effectively a counting problem that determines how many ways the term can contribute to the overall expansion. This problem requires an understanding of not only the Binomial Theorem but also the ability to manipulate exponents and coefficients systematically to isolate the desired term. Problems like these often serve as an introduction to more advanced topics such as generating functions or advanced combinatorial identities and are a staple in discrete mathematics courses.