Find the coefficient of x^4 in the expansion of (x - 1)^5.

Find the coefficient of x^4 in the expansion of (x - 1)^5.

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 to find the term that contains x^4. This occurs when n-k = 4, which means k = 1 (since 5 - 4 = 1).
Step 5: Calculate the binomial coefficient C(5, 1), which is the number of ways to choose 1 from 5. C(5, 1) = 5.
Step 6: The term we are interested in is C(5, 1) * (x)^(5-1) * (-1)^1.
Step 7: Substitute the values: 5 * x^4 * (-1).
Step 8: The coefficient of x^4 is 5 * (-1) = -5.

Binomial Expansion – The question tests the understanding of the binomial theorem, specifically how to find specific coefficients in the expansion of a binomial expression.
Combinatorial Coefficients – It involves calculating binomial coefficients, which represent the number of ways to choose elements from a set.
Negative Exponents – The solution requires understanding how to handle negative terms in the expansion.