Calculate the coefficient of x^2 in the expansion of (x + 1/2)^6.

Calculate the coefficient of x^2 in the expansion of (x + 1/2)^6.

Step 1: Identify the expression to expand, which is (x + 1/2)^6.
Step 2: Recognize that we need to find the coefficient of x^2 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 = 1/2, and n = 6.
Step 5: We want the term where x is raised to the power of 2, which means we need k = 4 (since n - k = 2).
Step 6: Calculate C(6, 4), which is the number of ways to choose 4 items from 6. This is equal to C(6, 2) because C(n, k) = C(n, n-k).
Step 7: C(6, 2) = 6! / (2! * (6-2)!) = (6 * 5) / (2 * 1) = 15.
Step 8: Now calculate (1/2)^4, which is (1/2) * (1/2) * (1/2) * (1/2) = 1/16.
Step 9: Multiply the coefficient C(6, 2) by (1/2)^4: 15 * (1/16) = 15/16.
Step 10: The coefficient of x^2 in the expansion of (x + 1/2)^6 is 15/16.

Binomial Expansion – The question tests the understanding of the binomial theorem, specifically how to find coefficients in the expansion of a binomial expression.
Combinatorics – It involves calculating combinations, as the coefficient of a term in the expansion is determined using binomial coefficients.
Powers of Fractions – The question requires knowledge of how to handle powers of fractions when calculating the coefficient.