What is the coefficient of x^4 in the expansion of (3x + 2)^6?

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Question: What is the coefficient of x^4 in the expansion of (3x + 2)^6?

Options:

Correct Answer: 720

Solution:

The coefficient of x^4 is C(6,4) * (3)^4 * (2)^2 = 15 * 81 * 4 = 4860.

What is the coefficient of x^4 in the expansion of (3x + 2)^6?

What is 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: Recognize that we need to find the coefficient of x^4 in this 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 = 3x, b = 2, and n = 6.
Step 5: We want the term where the power of x is 4. This happens when we take (3x)^4 and (2)^2.
Step 6: Calculate the number of ways to choose 4 from 6, which is C(6, 4). C(6, 4) = 15.
Step 7: Calculate (3)^4, which is 81.
Step 8: Calculate (2)^2, which is 4.
Step 9: Multiply these values together: C(6, 4) * (3)^4 * (2)^2 = 15 * 81 * 4.
Step 10: Perform the multiplication: 15 * 81 = 1215, then 1215 * 4 = 4860.
Step 11: Conclude that the coefficient of x^4 in the expansion of (3x + 2)^6 is 4860.

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