Final answer:
Using the binomial theorem, the coefficient of x^6y^1 in the expansion of (2x-2)^7 is found to be -896.
Step-by-step explanation:
The question involves finding the coefficient of x^6y^1 in the expansion of (2x-2)^7. To solve this, one can use the binomial theorem which states that the expansion of (a+b)^n will have terms of the form T(k+1) = (nCk) * a^(n-k) * b^k. Here, to find the term containing x^6y^1, we are looking for the term with x raised to the sixth power (which means k = 1, since the y is raised to the first power and also corresponds to the -2 term in our binomial).
First, we find the binomial coefficient using 7C1 (since we want the second term, and k=1). The binomial coefficient 7C1 is 7. Then, we raise 2x to the power of 6 and -2 to the power of 1 and multiply them with the binomial coefficient. Thus, 7 * (2x)^6 * (-2)^1 = 7 * 64x^6 * (-2) = -896x^6. Hence, the coefficient of x^6y^1 in the expansion is -896.