What is the coefficient of x^4 in the expansion of (x + 2)^6?

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Question: What is the coefficient of x^4 in the expansion of (x + 2)^6?

Options:

Correct Answer: 90

Solution:

The coefficient of x^4 is given by 6C4 * (2)^2 = 15 * 4 = 60.

What is the coefficient of x^4 in the expansion of (x + 2)^6?

What is the coefficient of x^4 in the expansion of (x + 2)^6?

Step 1: Identify the expression we need to expand, which is (x + 2)^6.
Step 2: Recognize that we want the coefficient of x^4 in this expansion.
Step 3: Use the binomial theorem, which states that (a + b)^n = sum from k=0 to n of (nCk * a^(n-k) * b^k).
Step 4: In our case, a = x, b = 2, and n = 6.
Step 5: We need to find the term where x is raised to the power of 4, which means we need k = 2 (since 6 - k = 4).
Step 6: Calculate the binomial coefficient 6C2, which is the number of ways to choose 2 from 6.
Step 7: 6C2 = 6! / (2! * (6-2)!) = 6! / (2! * 4!) = (6 * 5) / (2 * 1) = 15.
Step 8: Now, calculate (2)^2 because we have 2 raised to the power of k, which is 2.
Step 9: (2)^2 = 4.
Step 10: Multiply the coefficient from Step 7 and the result from Step 9: 15 * 4 = 60.
Step 11: Therefore, the coefficient of x^4 in the expansion of (x + 2)^6 is 60.

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