Find the coefficient of x^4 in the expansion of (x + 5)^7.

Find the coefficient of x^4 in the expansion of (x + 5)^7.

Step 1: Identify the expression we need to expand, which is (x + 5)^7.
Step 2: Understand 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 = 7.
Step 5: We need to find the term where x is raised to the power of 4, which means we need k = 3 (since 7 - 4 = 3).
Step 6: Calculate C(7, 3), which is the number of ways to choose 3 items from 7. This is calculated as 7! / (3! * (7-3)!) = 35.
Step 7: Calculate 5^3, which is 5 * 5 * 5 = 125.
Step 8: Multiply the results from Step 6 and Step 7 to find the coefficient: 35 * 125 = 4375.

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 the coefficient.
Power of a Constant – Calculating the power of a constant (in this case, 5) raised to the appropriate exponent is necessary to find the coefficient of the desired term.