What is the coefficient of x^3 in the expansion of (x - 1)^5? (2022)

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Question: What is the coefficient of x^3 in the expansion of (x - 1)^5? (2022)

Options:

Correct Answer: -10

Solution:

The coefficient of x^3 is C(5,3)(-1)^2 = 10.

What is the coefficient of x^3 in the expansion of (x - 1)^5? (2022)

What is the coefficient of x^3 in the expansion of (x - 1)^5? (2022)

Step 1: Identify the expression we need to expand, which is (x - 1)^5.
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 = 5.
Step 4: We want the term where x is raised to the power of 3, which means we need to find the term where k = 2 (since n - k = 3).
Step 5: Calculate C(5, 2), which is the number of ways to choose 2 items from 5. This is equal to 5! / (2! * (5-2)!) = 10.
Step 6: The term we are interested in is C(5, 2) * (x)^(5-2) * (-1)^2.
Step 7: Substitute the values: 10 * x^3 * 1 = 10 * x^3.
Step 8: The coefficient of x^3 in the expansion is 10.

Binomial Expansion – The question tests the understanding of the binomial theorem, specifically how to find specific coefficients in the expansion of a binomial expression.
Combinatorics – The use of combinations (C(n, k)) to determine the number of ways to choose k successes in n trials is essential for calculating coefficients.
Negative Exponents – Understanding how to handle negative terms in the binomial expansion, particularly when raising a negative number to a power.