Coefficient of Third Term in Binomial Expansion

Determine the coefficient of the third term in the binomial expansion of $(2x + y)^5$.

The binomial expansion is an important tool in discrete mathematics, especially when dealing with polynomial expressions raised to a power. When expanding a binomial expression such as $(2x + y)$ raised to the fifth power, we use the Binomial Theorem. This theorem provides a way to expand expressions of the form $(a + b)^n$, and it states that each term in the expansion is of the form $(n \choose k) \cdot a^{(n-k)} \cdot b^k$, where "n choose k" denotes a binomial coefficient.

The concept of binomial coefficients is central here. These coefficients can be found in Pascal's Triangle or calculated with the formula $\frac{n!}{k!(n-k)!}$, where $n!$ represents the factorial of n. Recognizing that the third term in the expansion corresponds to $k = 2$, we apply these principles to find the coefficient. Familiarity with concepts like factorials and combinations (ways to choose k elements from n without regard to order) will be advantageous.

Understanding these ideas not only aids in directly solving problems about expansions but also extends to other areas of discrete mathematics, such as probability and combinatorics, where these coefficients and expansions are used frequently. This problem offers a chance to explore the connection between algebraic identities and combinatorial reasoning, enhancing one's ability to tackle a diverse range of mathematical problems.