Calculate the coefficient of x^3 in the expansion of (x - 1)^5.

Calculate the coefficient of x^3 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^3. This happens when n-k = 3, which means k = 5 - 3 = 2.
Step 5: Calculate C(5, 2), which is the number of ways to choose 2 items from 5. C(5, 2) = 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 = 10x^3.
Step 8: The coefficient of x^3 in the expansion is 10.

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.