Find the coefficient of x^4 in the expansion of (2x - 3)^5.

Find the coefficient of x^4 in the expansion of (2x - 3)^5.

Step 1: Identify the expression to expand, which is (2x - 3)^5.
Step 2: Use the Binomial Theorem, which states that (a + b)^n = Σ [C(n, k) * a^(n-k) * b^k] for k = 0 to n.
Step 3: In our case, a = 2x, b = -3, and n = 5.
Step 4: We want the term where the power of x is 4, which means we need to find the term where k = 1 (since 5 - k = 4).
Step 5: Calculate C(5, 1), which is the number of ways to choose 1 from 5. C(5, 1) = 5.
Step 6: Calculate (2x)^(5-1) = (2x)^4 = 2^4 * x^4 = 16 * x^4.
Step 7: Calculate (-3)^1 = -3.
Step 8: Combine these results: Coefficient = C(5, 1) * (2^4) * (-3) = 5 * 16 * (-3).
Step 9: Calculate the final result: 5 * 16 = 80, then 80 * (-3) = -240.
Step 10: The coefficient of x^4 in the expansion is -240.

Binomial Expansion – The question tests the understanding of the binomial theorem, specifically how to find specific coefficients in the expansion of a binomial expression.
Combinatorics – It involves calculating combinations, as represented by C(5,4), which is essential for determining the number of ways to choose terms in the expansion.
Exponentiation – The question requires knowledge of how to handle powers of numbers, particularly when multiplying coefficients.