In the expansion of (2x + 3)^6, what is the coefficient of x^4?

In the expansion of (2x + 3)^6, what is the coefficient of x^4?

Correct Answer: 2160

Step 1: Identify the expression to expand, which is (2x + 3)^6.
Step 2: Recognize that we need to find the coefficient of x^4 in the expansion.
Step 3: 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 4: In our case, a = 2x, b = 3, and n = 6.
Step 5: We want the term where x has the power of 4, which means we need (2x)^(4) and (3)^(2).
Step 6: To find this term, we set k = 2 (because 6 - 4 = 2).
Step 7: Calculate C(6, 2), which is the number of ways to choose 2 from 6. C(6, 2) = 6! / (2!(6-2)!) = 15.
Step 8: Calculate (2)^4, which is 16.
Step 9: Calculate (3)^2, which is 9.
Step 10: Multiply these values together: Coefficient = C(6, 2) * (2)^4 * (3)^2 = 15 * 16 * 9.
Step 11: Perform the multiplication: 15 * 16 = 240, then 240 * 9 = 2160.
Step 12: Conclude that the coefficient of x^4 in the expansion of (2x + 3)^6 is 2160.

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