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

Find 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 - 5 = 2).
Step 6: Calculate C(7, 2), which is the number of ways to choose 2 items 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 of x^5: 21 * 4 = 84.

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 – Understanding how to calculate the power of a constant (in this case, 2) when determining the coefficient of a specific term in the expansion.