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

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

Step 1: Identify the expression to expand, which is (3x - 4)^4.
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 = -4, and n = 4.
Step 4: We want the coefficient of x^2, which means we need the term where (3x) is raised to the power of 2.
Step 5: This occurs when k = 2, because (3x)^(2) corresponds to x^2.
Step 6: Calculate C(4, 2), which is the number of ways to choose 2 from 4. C(4, 2) = 4! / (2! * (4-2)!) = 6.
Step 7: Calculate (3)^2, which is 9.
Step 8: Calculate (-4)^2, which is 16.
Step 9: Multiply these values together: Coefficient = C(4, 2) * (3)^2 * (-4)^2 = 6 * 9 * 16.
Step 10: Perform the multiplication: 6 * 9 = 54, then 54 * 16 = 864.
Step 11: The coefficient of x^2 in the expansion of (3x - 4)^4 is 864.

Binomial Expansion – The question tests the understanding of the binomial theorem, specifically how to find coefficients in the expansion of a binomial expression.
Combinatorics – It involves calculating combinations, specifically C(4,2), which represents the number of ways to choose 2 terms from 4.
Exponentiation – The question requires knowledge of how to handle powers of numbers, particularly when applying them in the binomial expansion.