Calculate the coefficient of x^2 in the expansion of (x + 4)^6.

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Question: Calculate the coefficient of x^2 in the expansion of (x + 4)^6.

Options:

Correct Answer: 96

Solution:

The coefficient of x^2 is given by C(6,2) * 4^4 = 15 * 256 = 3840.

Calculate the coefficient of x^2 in the expansion of (x + 4)^6.

Calculate the coefficient of x^2 in the expansion of (x + 4)^6.

Step 1: Identify the expression to expand, which is (x + 4)^6.
Step 2: Understand that we want the coefficient of x^2 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 = 4, and n = 6.
Step 5: We need to find the term where x is raised to the power of 2, which means we need k = 4 (since 6 - 2 = 4).
Step 6: Calculate C(6, 2), which is the number of ways to choose 2 items from 6. This is calculated as 6! / (2!(6-2)!) = 15.
Step 7: Calculate 4^4, which is 4 * 4 * 4 * 4 = 256.
Step 8: Multiply the results from Step 6 and Step 7: 15 * 256 = 3840.
Step 9: The coefficient of x^2 in the expansion of (x + 4)^6 is 3840.

Binomial Expansion – The process of expanding expressions of the form (a + b)^n using the binomial theorem.
Binomial Coefficient – The coefficient C(n, k) represents the number of ways to choose k elements from a set of n elements.
Power of a Constant – Calculating powers of constants, such as 4^4 in this case.