Find the coefficient of x^3 in the expansion of (x - 1)^6.

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Question: Find the coefficient of x^3 in the expansion of (x - 1)^6.

Options:

Correct Answer: -20

Solution:

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

Find the coefficient of x^3 in the expansion of (x - 1)^6.

Find the coefficient of x^3 in the expansion of (x - 1)^6.

Correct Answer: -20

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 to find the term where the power of x is 3, which means we need to find the term where k = 3 (since x^(6-k) = x^3 when k = 3).
Step 5: 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 6: The term we are interested in is C(6, 3) * (x)^(6-3) * (-1)^3.
Step 7: Substitute the values: C(6, 3) = 20, (x)^(6-3) = x^3, and (-1)^3 = -1.
Step 8: Combine these to find the term: 20 * x^3 * (-1) = -20 * x^3.
Step 9: The coefficient of x^3 in the expansion is -20.

Binomial Expansion – The process of expanding expressions of the form (a + b)^n using the binomial theorem.
Coefficients in Binomial Expansion – Understanding how to find specific coefficients in the expansion using combinations.
Negative Exponents – Recognizing the effect of negative terms in the expansion.