Find the coefficient of x^4 in the expansion of (x - 5)^6.

Find the coefficient of x^4 in the expansion of (x - 5)^6.

Step 1: Identify the expression we need to expand, which is (x - 5)^6.
Step 2: Recognize that we want to find the coefficient of x^4 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 = -5, and n = 6.
Step 5: We need to find the term where the power of x is 4. This means we want k = 2 because 6 - k = 4.
Step 6: Calculate C(6, 2), which is the number of ways to choose 2 items from 6. C(6, 2) = 6! / (2!(6-2)!) = 15.
Step 7: Calculate (-5)^2, which is 25.
Step 8: Multiply the results from Step 6 and Step 7: 15 * 25 = 375.
Step 9: Conclude that the coefficient of x^4 in the expansion of (x - 5)^6 is 375.

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 the combination formula C(n, k) to determine the number of ways to choose k successes in n trials is essential for finding specific coefficients in the expansion.
Negative Coefficients – Understanding how to handle negative coefficients in the expansion, particularly when raised to a power.