Calculate the coefficient of x^5 in the expansion of (x + 2)^7.

Calculate the coefficient of x^5 in the expansion of (x + 2)^7.

Step 1: Identify the expression we need to expand, which is (x + 2)^7.
Step 2: Understand that we want to find the coefficient of x^5 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 = 7.
Step 5: We need to find the term where x is raised to the power of 5. This means we need k = 2 because 7 - k = 5.
Step 6: Calculate C(7, 2), which is the number of ways to choose 2 from 7. C(7, 2) = 7! / (2!(7-2)!) = 21.
Step 7: Calculate (2)^2, which is the value of b raised to the power of k. (2)^2 = 4.
Step 8: Multiply the results from Step 6 and Step 7 to find the coefficient: 21 * 4 = 84.

Binomial Expansion – The process of expanding expressions of the form (a + b)^n using the binomial theorem.
Binomial Coefficient – The coefficient C(n, k) represents the number of ways to choose k elements from a set of n elements.
Power of a Term – Understanding how to determine the power of each term in the expansion and how it contributes to the overall expression.