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

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

Options:

Correct Answer: 243

Solution:

The coefficient of x^4 in (x + 3)^6 is C(6, 4) * 3^2 = 15 * 9 = 135.

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

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

Step 1: Identify the expression we need to expand, which is (x + 3)^6.
Step 2: Recognize that we want 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 = x, b = 3, and n = 6.
Step 5: We need to find the term where x is raised to the power of 4, which means we need k = 2 (since 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 3^2, which is the value of b raised to the power of k. 3^2 = 9.
Step 8: Multiply the results from Step 6 and Step 7 to find the coefficient: 15 * 9 = 135.

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