What is the coefficient of x^4 in the expansion of (2x + 5)^6? (2022)

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Question: What is the coefficient of x^4 in the expansion of (2x + 5)^6? (2022)

Options:

Correct Answer: 450

Exam Year: 2022

Solution:

The coefficient of x^4 is C(6,4) * (2)^4 * (5)^2 = 15 * 16 * 25 = 600.

What is the coefficient of x^4 in the expansion of (2x + 5)^6? (2022)

What is the coefficient of x^4 in the expansion of (2x + 5)^6? (2022)

Step 1: Identify the expression to expand, which is (2x + 5)^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 = 2x, b = 5, and n = 6.
Step 5: We want the term where the power of x is 4, which means we need (2x)^4 and (5)^(6-4) = (5)^2.
Step 6: Calculate the coefficient using C(6, 4), which is the number of ways to choose 4 from 6.
Step 7: C(6, 4) = 6! / (4! * (6-4)!) = 15.
Step 8: Calculate (2)^4 = 16.
Step 9: Calculate (5)^2 = 25.
Step 10: Multiply the results: Coefficient = C(6, 4) * (2)^4 * (5)^2 = 15 * 16 * 25.
Step 11: Perform the multiplication: 15 * 16 = 240, then 240 * 25 = 600.
Step 12: Conclude that the coefficient of x^4 in the expansion of (2x + 5)^6 is 600.

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