What is the value of the coefficient of x^5 in the expansion of (x + 2)^7?

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Question: What is the value of the coefficient of x^5 in the expansion of (x + 2)^7?

Options:

Correct Answer: 56

Solution:

The coefficient of x^5 is given by 7C5 * (2)^2 = 21 * 4 = 84.

What is the value of the coefficient of x^5 in the expansion of (x + 2)^7?

What is the value of the coefficient of x^5 in the expansion of (x + 2)^7?

Step 1: Identify the expression we are expanding, which is (x + 2)^7.
Step 2: We need to find the coefficient of x^5 in this expansion.
Step 3: Use the binomial theorem, which states that (a + b)^n = sum of (nCk * a^(n-k) * b^k) for k from 0 to n.
Step 4: In our case, a = x, b = 2, and n = 7.
Step 5: We want the term where x is raised to the power of 5, which means we need k = 2 (because 7 - 5 = 2).
Step 6: Calculate the binomial coefficient 7C2, which is the number of ways to choose 2 items from 7. This is calculated as 7! / (2! * (7-2)!) = 21.
Step 7: Now, calculate (2)^2, which is 4.
Step 8: Multiply the binomial coefficient by (2)^2: 21 * 4 = 84.
Step 9: The coefficient of x^5 in the expansion of (x + 2)^7 is 84.

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