What is the coefficient of x^6 in the expansion of (x + 3)^9? (2020)

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Question: What is the coefficient of x^6 in the expansion of (x + 3)^9? (2020)

Options:

Correct Answer: 126

Solution:

The coefficient of x^6 is C(9,6)(3)^3 = 84 * 27 = 2268.

What is the coefficient of x^6 in the expansion of (x + 3)^9? (2020)

What is the coefficient of x^6 in the expansion of (x + 3)^9? (2020)

Step 1: Identify the expression we need to expand, which is (x + 3)^9.
Step 2: Recognize that we want the coefficient of x^6 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 = 9.
Step 5: We need to find the term where x is raised to the power of 6, which means we need k = 3 (since 9 - 6 = 3).
Step 6: Calculate the binomial coefficient C(9, 3), which is the number of ways to choose 3 items from 9.
Step 7: C(9, 3) = 9! / (3! * (9-3)!) = 9! / (3! * 6!) = (9 * 8 * 7) / (3 * 2 * 1) = 84.
Step 8: Now calculate 3^3, which is 3 * 3 * 3 = 27.
Step 9: Multiply the coefficient C(9, 3) by 3^3 to find the coefficient of x^6: 84 * 27.
Step 10: Calculate 84 * 27 = 2268.

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