What is the value of the coefficient of x^4 in the expansion of (x - 2)^6? (2023

What is the value of the coefficient of x^4 in the expansion of (x - 2)^6? (2023)

What is the value of the coefficient of x^4 in the expansion of (x - 2)^6? (2023)

Step 1: Identify the expression we need to expand, which is (x - 2)^6.
Step 2: 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 3: In our case, a = x, b = -2, and n = 6.
Step 4: We want the coefficient of x^4, which means we need to find the term where the power of x is 4.
Step 5: To find this term, we set n-k = 4, which means k = 6 - 4 = 2.
Step 6: Calculate the binomial coefficient C(6, 2), which is the number of ways to choose 2 from 6.
Step 7: C(6, 2) = 6! / (2! * (6-2)!) = (6 * 5) / (2 * 1) = 15.
Step 8: Now, we need to calculate (-2)^k, where k = 2. So, (-2)^2 = 4.
Step 9: Multiply the binomial coefficient by (-2)^2: 15 * 4 = 60.
Step 10: Since we are looking for the coefficient of x^4 in (x - 2)^6, we need to consider the sign. The term is negative because of (-2), so the coefficient is -60.

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