Expand Using Pascals Triangle

Expand $(2x-3y)^5$ using Pascal's triangle.

In this exercise, we explore the application of Pascal's triangle to the binomial expansion of expressions. The problem involves expanding a binomial raised to a power, a fundamental concept in combinatorics and algebra. Pascal's triangle provides a straightforward way of determining the coefficients in the expansion of a binomial expression. The rows of Pascal's triangle correspond to the coefficients of the binomial expansions, which means that for a binomial raised to the fifth power, we look at the sixth row of Pascal's triangle, as the first row corresponds to the power of zero.

By using Pascal's triangle, we can determine the coefficients for each term in the expansion of $(2x - 3y)^5$. Each term of the expanded expression is formed by taking the relevant binomial coefficient, multiplying it by the appropriate powers of the two terms of the binomial expression, and then combining them. This technique illustrates the idea of selecting terms from a set and is foundational for understanding more complex topics in discrete mathematics, such as combinatorial identities and multinomial expansions. Understanding how to expand binomials using Pascal's triangle is not only useful for solving mathematical problems, but it also provides a deeper insight into the nature of polynomial functions and their applications in computer science, such as algorithm design and analysis.