What is the coefficient of x^6 in the expansion of (x + 3)^8? (2020)

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Question: What is the coefficient of x^6 in the expansion of (x + 3)^8? (2020)

Options:

Correct Answer: 84

Exam Year: 2020

Solution:

The coefficient of x^6 is C(8,6) * (3)^2 = 28 * 9 = 252.

What is the coefficient of x^6 in the expansion of (x + 3)^8? (2020)

What is the coefficient of x^6 in the expansion of (x + 3)^8? (2020)

Step 1: Identify the expression we need to expand, which is (x + 3)^8.
Step 2: Understand 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 = 8.
Step 5: We need to find the term where x is raised to the power of 6. This means we need k = 2 (since 8 - 6 = 2).
Step 6: Calculate C(8, 2), which is the number of ways to choose 2 items from 8. C(8, 2) = 8! / (2!(8-2)!) = 28.
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 of x^6: 28 * 9 = 252.

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