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

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

Correct Answer: 2.5

Step 1: Identify the expression to expand, which is (x + 1/2)^6.
Step 2: 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 3: In our case, a = x, b = 1/2, and n = 6.
Step 4: We want the coefficient of x^3, which means we need to find the term where k = 3 (since x is raised to the power of 3).
Step 5: Calculate C(6, 3), which is the number of ways to choose 3 items from 6. C(6, 3) = 6! / (3! * (6-3)!) = 20.
Step 6: Calculate (1/2)^3, which is 1/2 * 1/2 * 1/2 = 1/8.
Step 7: Multiply the coefficient C(6, 3) by (1/2)^3: 20 * (1/8) = 20/8 = 2.5.
Step 8: 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 coefficients in the expansion of a binomial expression.
Combinatorics – It involves calculating combinations, represented as C(n, k), which is crucial for determining the coefficients in the expansion.
Powers of Fractions – The question requires knowledge of how to handle powers of fractions, particularly when calculating (1/2)^3.