What is the coefficient of x^5 in the expansion of (2x - 1)^6?

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Question: What is the coefficient of x^5 in the expansion of (2x - 1)^6?

Options:

Correct Answer: 80

Solution:

The coefficient of x^5 is C(6,5) * (2)^5 * (-1)^1 = 6 * 32 * (-1) = -192.

What is the coefficient of x^5 in the expansion of (2x - 1)^6?

What is the coefficient of x^5 in the expansion of (2x - 1)^6?

Step 1: Identify the expression we need to expand, which is (2x - 1)^6.
Step 2: Use the Binomial Theorem, which states that (a + b)^n = sum(C(n, k) * a^(n-k) * b^k) for k from 0 to n.
Step 3: In our case, a = 2x, b = -1, and n = 6.
Step 4: We want the term that contains x^5. This occurs when we choose (2x) 5 times and (-1) 1 time.
Step 5: Calculate the number of ways to choose 5 from 6, which is C(6, 5). C(6, 5) = 6.
Step 6: Calculate (2x)^5. This is (2^5)(x^5) = 32x^5.
Step 7: Calculate (-1)^1, which is -1.
Step 8: Combine these results to find the coefficient: Coefficient = C(6, 5) * (2^5) * (-1) = 6 * 32 * (-1).
Step 9: Calculate 6 * 32 = 192.
Step 10: Multiply by -1 to get the final coefficient: -192.

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