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We are given the binomial expansion of \((2x - 2y)^6\), and asked to find the sixth term. To approach this, we will use the binomial theorem, which gives the general term of the expansion as:

\[
T_{k+1} = \binom{n}{k} a^{n-k} b^k
\]

Here:
- \( n = 6 \)
- \( a = 2x \)
- \( b = -2y \)
- The term we need is the 6th term, so \( k = 5 \).

### Step-by-Step Process:

1. **Apply the binomial coefficient:**
   \[
   \binom{6}{5} = \frac{6!}{5!(6-5)!} = 6
   \]

2. **Substitute into the general term formula:**
   \[
   T_6 = \binom{6}{5} (2x)^{6-5} (-2y)^5
   \]
   Simplifying the exponents:
   \[
   T_6 = 6 \cdot (2x)^1 \cdot (-2y)^5
   \]

3. **Evaluate the powers:**
   \[
   (2x)^1 = 2x
   \]
   \[
   (-2y)^5 = -32y^5
   \]

4. **Multiply everything together:**
   \[
   T_6 = 6 \cdot 2x \cdot (-32y^5) = 6 \cdot 2 \cdot (-32) \cdot xy^5
   \]
   \[
   T_6 = -384xy^5
   \]

### Final Answer:
The sixth term is \(-384xy^5\), which corresponds to the first option in the choices.

Would you like a detailed breakdown of another part or have any questions?

Here are five related questions to explore:
1. How is the binomial coefficient calculated in general?
2. What would be the seventh term of this binomial expansion?
3. How does the sign change when expanding negative terms like \((-2y)\)?
4. What is the significance of Pascal's Triangle in binomial expansion?
5. Can you generalize this approach for any binomial expansion?

**Tip:** When expanding binomials with negative terms, pay close attention to the powers of the negative number, as they alternate the signs of the terms!

What is the sixth term of the binomial expansion for (2x - 2y)^6?

Learn how to find the sixth term in the binomial expansion of (2x - 2y)^6 using the binomial theorem. This detailed solution covers key algebraic concepts, including the use of binomial coefficients and powers. Suitable for Grades 10-12.

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