Find the coefficient of x^3 in the expansion of (x - 1)^7.

Find the coefficient of x^3 in the expansion of (x - 1)^7.

Step 1: Identify the expression we need to expand, which is (x - 1)^7.
Step 2: Use the Binomial Theorem, which states that (a + b)^n = sum of (nCk * a^(n-k) * b^k) for k from 0 to n.
Step 3: In our case, a = x, b = -1, and n = 7.
Step 4: We want to find the term that contains x^3. This happens when n-k = 3, which means k = 7 - 3 = 4.
Step 5: Calculate the binomial coefficient for k = 4, which is 7C4.
Step 6: The formula for the binomial coefficient is nCk = n! / (k! * (n-k)!). So, 7C4 = 7! / (4! * 3!) = 35.
Step 7: The term we are interested in is 7C4 * (x)^(7-4) * (-1)^4.
Step 8: Substitute the values: 35 * x^3 * 1 = 35 * x^3.
Step 9: The coefficient of x^3 in the expansion is 35.

Binomial Expansion – The question tests the understanding of the binomial theorem, which allows for the expansion of expressions of the form (a + b)^n.
Combination Formula – The use of combinations (nCr) to determine the number of ways to choose terms from the expansion is essential for finding specific coefficients.
Sign of Terms – Understanding how to handle negative signs in the expansion is crucial, as it affects the coefficient calculation.