Find the coefficient of x^3 in the expansion of (x + 2)^6.

Find the coefficient of x^3 in the expansion of (x + 2)^6.

Step 1: Identify the expression we need to expand, which is (x + 2)^6.
Step 2: Understand that we want to find the coefficient of x^3 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 = 2, and n = 6.
Step 5: We need to find the term where x is raised to the power of 3, which means we need k = 3 (since n - k = 3).
Step 6: Calculate C(6, 3), which is the number of ways to choose 3 items from 6. This is calculated as 6! / (3! * (6-3)!) = 20.
Step 7: Calculate (2)^3, which is 2 * 2 * 2 = 8.
Step 8: Multiply the results from Step 6 and Step 7 to find the coefficient: 20 * 8 = 160.

Binomial Expansion – The process of expanding expressions of the form (a + b)^n using the binomial theorem.
Combination Formula – Understanding how to calculate combinations, denoted as C(n, k), which represents the number of ways to choose k elements from a set of n elements.
Coefficients in Polynomial Expansion – Identifying and calculating the coefficients of specific terms in the expansion of polynomials.