What is the value of the coefficient of x^3 in the expansion of (x - 1)^6?

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Question: What is the value of the coefficient of x^3 in the expansion of (x - 1)^6?

Options:

Correct Answer: -30

Solution:

The coefficient of x^3 is C(6,3) * (-1)^3 = 20 * (-1) = -20.

What is the value of the coefficient of x^3 in the expansion of (x - 1)^6?

What is the value of the coefficient of x^3 in the expansion of (x - 1)^6?

Step 1: Identify the expression we need to expand, which is (x - 1)^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 = -1, and n = 6.
Step 4: We want the coefficient of x^3, so we need to find the term where the power of x is 3.
Step 5: This occurs when k = 3, because x^(6-k) = x^(6-3) = x^3.
Step 6: Calculate C(6, 3), which is the number of ways to choose 3 items from 6. C(6, 3) = 6! / (3! * (6-3)!) = 20.
Step 7: The term we are interested in is C(6, 3) * x^3 * (-1)^3.
Step 8: Since (-1)^3 = -1, the term becomes 20 * x^3 * (-1) = -20 * x^3.
Step 9: Therefore, the coefficient of x^3 in the expansion is -20.

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