Find the coefficient of x^4 in the expansion of (2x - 3)^6.

Find the coefficient of x^4 in the expansion of (2x - 3)^6.

Step 1: Identify the expression to expand, which is (2x - 3)^6.
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 = 6.
Step 4: We want the term where the power of x is 4, which means we need (2x) raised to the power of 4.
Step 5: This corresponds to k = 2 in the Binomial Theorem because n - k = 4, so k = 6 - 4 = 2.
Step 6: Calculate C(6, 2), which is the number of ways to choose 2 from 6. C(6, 2) = 6! / (2!(6-2)!) = 15.
Step 7: Calculate (2x)^4, which is (2^4)(x^4) = 16x^4.
Step 8: Calculate (-3)^2, which is 9.
Step 9: Multiply the results: Coefficient = C(6, 2) * (2^4) * (-3)^2 = 15 * 16 * 9.
Step 10: Calculate 15 * 16 = 240, then 240 * 9 = 2160.
Step 11: The coefficient of x^4 in the expansion is 2160.

Binomial Theorem – The Binomial Theorem is used to expand expressions of the form (a + b)^n, where the coefficients can be calculated using combinations.
Combination Formula – The combination formula C(n, k) = n! / (k!(n-k)!) is used to determine the number of ways to choose k elements from a set of n elements.
Power of a Product – When raising a product to a power, each factor in the product is raised to that power.