What is the value of the coefficient of x^5 in the expansion of (x + 1/2)^8?

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Question: What is the value of the coefficient of x^5 in the expansion of (x + 1/2)^8?

Options:

Correct Answer: 56

Solution:

The coefficient of x^5 is given by 8C5 * (1/2)^3 = 56 * 1/8 = 7.

What is the value of the coefficient of x^5 in the expansion of (x + 1/2)^8?

What is the value of the coefficient of x^5 in the expansion of (x + 1/2)^8?

Step 1: Identify the expression we are expanding, which is (x + 1/2)^8.
Step 2: Recognize that we need to find the coefficient of x^5 in this expansion.
Step 3: Use the binomial theorem, which states that (a + b)^n = Σ (nCk * a^(n-k) * b^k) for k = 0 to n.
Step 4: In our case, a = x, b = 1/2, and n = 8.
Step 5: We want the term where x is raised to the power of 5, which means we need k = 3 (since 8 - 5 = 3).
Step 6: Calculate the binomial coefficient 8C3, which is the number of ways to choose 3 items from 8. This is calculated as 8! / (3! * (8-3)!) = 56.
Step 7: Calculate (1/2)^3, which is 1/2 * 1/2 * 1/2 = 1/8.
Step 8: Multiply the binomial coefficient by (1/2)^3 to find the coefficient of x^5: 56 * (1/8) = 7.

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