Calculate the coefficient of x^4 in the expansion of (x + 1/2)^6. (2021)

Calculate the coefficient of x^4 in the expansion of (x + 1/2)^6. (2021)

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^4, which means we need to find the term where x is raised to the power of 4.
Step 5: For x^4, we have n-k = 4, so k = 6 - 4 = 2.
Step 6: Calculate the binomial coefficient C(6, 2), which is the number of ways to choose 2 from 6.
Step 7: C(6, 2) = 6! / (2! * (6-2)!) = (6 * 5) / (2 * 1) = 15.
Step 8: Now, calculate (1/2)^k, where k = 2. This gives us (1/2)^2 = 1/4.
Step 9: Multiply the coefficient from Step 7 by the result from Step 8: 15 * (1/4) = 15/4.
Step 10: The coefficient of x^4 in the expansion of (x + 1/2)^6 is 15/4.

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 – The use of combinations (C(n, k)) to determine the number of ways to choose k successes in n trials is essential for calculating the coefficient.
Powers of Fractions – Understanding how to handle fractional powers when calculating coefficients in the expansion.