Calculate the coefficient of x^3 in the expansion of (x + 1/2)^6.

Calculate the coefficient of x^3 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^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 = 1/2, and n = 6.
Step 5: We want 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 (1/2)^3, which is 1/2 * 1/2 * 1/2 = 1/8.
Step 8: Multiply the results from Step 6 and Step 7: 20 * (1/8) = 20/8 = 2.5.
Step 9: Conclude that the coefficient of x^3 in the expansion of (x + 1/2)^6 is 2.5.

Binomial Expansion – The question tests the understanding of the binomial theorem, specifically how to find specific coefficients in the expansion of a binomial expression.
Combinatorics – It involves calculating combinations, specifically C(6,3), which represents the number of ways to choose 3 successes (x terms) from 6 trials.
Powers of Fractions – The question requires knowledge of how to handle powers of fractions, particularly (1/2)^3.