In the expansion of (x - 1)^5, what is the coefficient of x^3?

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Question: In the expansion of (x - 1)^5, what is the coefficient of x^3?

Options:

Correct Answer: -10

Solution:

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

In the expansion of (x - 1)^5, what is the coefficient of x^3?

In the expansion of (x - 1)^5, what is the coefficient of x^3?

Step 1: Identify the expression we are expanding, which is (x - 1)^5.
Step 2: Recognize that we need to find the coefficient of x^3 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 = -1, and n = 5.
Step 5: We want the term where x is raised to the power of 3, which means we need k = 5 - 3 = 2.
Step 6: Calculate the binomial coefficient C(5, 2), which is the number of ways to choose 2 items from 5.
Step 7: C(5, 2) = 5! / (2! * (5-2)!) = 10.
Step 8: Since b = -1, we need to consider (-1)^2 for the term corresponding to k = 2.
Step 9: Calculate (-1)^2, which equals 1.
Step 10: Multiply the binomial coefficient by (-1)^2: 10 * 1 = 10.
Step 11: Therefore, the coefficient of x^3 in the expansion of (x - 1)^5 is 10.

Binomial Expansion – Understanding how to expand expressions of the form (a + b)^n using the Binomial Theorem.
Coefficients in Binomial Expansion – Identifying and calculating the coefficients of specific terms in the expansion.
Combinatorial Notation – Using combinations, denoted as C(n, k), to find the number of ways to choose k elements from n elements.