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

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

Step 1: Identify the expression to expand, which is (3x - 2)^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 = 3x, b = -2, and n = 6.
Step 4: We want the coefficient of x^4, which means we need the term where (3x) is raised to the power of 4.
Step 5: This occurs when k = 2 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 (3)^4, which is 3 raised to the power of 4. (3)^4 = 81.
Step 8: Calculate (-2)^2, which is -2 raised to the power of 2. (-2)^2 = 4.
Step 9: Multiply the results: Coefficient = C(6, 2) * (3)^4 * (-2)^2 = 15 * 81 * 4.
Step 10: Calculate 15 * 81 = 1215, then multiply by 4 to get 4860.

Binomial Theorem – The Binomial Theorem provides a way to expand expressions of the form (a + b)^n, where the coefficients can be calculated using combinations.
Combinations – Understanding how to calculate combinations (C(n, k)) is essential for determining the coefficients in the binomial expansion.
Exponent Rules – Applying exponent rules correctly when calculating powers of terms in the expansion.