What is the value of the coefficient of x^4 in the expansion of (2x - 5)^6?

What is the value of the coefficient of x^4 in the expansion of (2x - 5)^6?

Step 1: Identify the expression we need to expand, which is (2x - 5)^6.
Step 2: 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 3: In our case, a = 2x, b = -5, and n = 6.
Step 4: We want the coefficient of x^4, which means we need to find the term where (2x) is raised to the power of 4.
Step 5: If (2x) is raised to the power of 4, then (-5) must be raised to the power of (6 - 4) = 2.
Step 6: The term we are looking for is given by the formula: nCk * (2x)^(n-k) * (-5)^k, where k = 2.
Step 7: Calculate nCk, which is 6C2. This is equal to 6! / (2!(6-2)!) = 15.
Step 8: Calculate (2)^4, which is 16.
Step 9: Calculate (-5)^2, which is 25.
Step 10: Multiply these values together: 15 * 16 * 25.
Step 11: Calculate 15 * 16 = 240.
Step 12: Then calculate 240 * 25 = 600.
Step 13: Since the term involves (-5), the coefficient will be negative, so the final coefficient is -600.

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