Find the coefficient of x
2y
4
in the binomial expansion of
x
3
− 6y
6
.

To find the coefficient of \( x^2y^4 \) in the binomial expansion of \( (x^3 - 6y)^6 \), we can use the binomial theorem.

The binomial theorem states:

\[
(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k
\]

Here, we have \( a = x^3 \), \( b = -6y \), and \( n = 6 \). We need to find the term in the expansion that contains \( x^2y^4 \).

In the general term of the expansion:

\[
\text{Term} = \binom{6}{k} (x^3)^{6-k} (-6y)^k
\]

Simplifying this term:

\[
\text{Term} = \binom{6}{k} x^{3(6-k)} (-6)^k y^k
\]

We want this term to be of the form \( x^2y^4 \), so we need to solve the following equations:

1. \( 3(6-k) = 2 \) for the power of \( x \)
2. \( k = 4 \) for the power of \( y \)

Let's solve these step by step:

### Step 1: Solve for \( k \) from the equation \( 3(6-k) = 2 \)
\[
3(6-k) = 2
\]
\[
18 - 3k = 2
\]
\[
3k = 16
\]
\[
k = \frac{16}{3}
\]

Since \( k \) must be an integer, there is no integer solution for this. This suggests a misinterpretation. Let's correct the approach by directly setting \( k = 4 \) (from the requirement \( y^4 \) and checking the power of \( x \)).

### Step 2: Verify the power of \( x \) with \( k = 4 \)
\[
\text{Power of } x = 3(6-4) = 3(2) = 6
\]

This does not match \( x^2 \). Therefore, \( k = 4 \) is incorrect.

### Step 3: Analyze the correct terms
Let's consider that no such direct term matches \( x^2y^4 \). The usual approach to identify such errors is to reconsider the degree, or to match it to another value.

Given this, it seems the problem might be misstated. Let's check a simplified approach to reflect on your exact requirement.

Could you clarify if the powers intended are true for \( y^4 \) specifically?

---

Meanwhile, some potential follow-up queries could be:
1. Any specific way you want these powers laid out?
2. Are there more similar terms?
3. Shall we refine it considering the whole?
4. How about recalculating the base terms?
5. Any specific limits to these powers?

**Tip**: Ensure checking powers directly relates to known binomial identities or derivations.

Learn how to find the coefficient of x^2y^4 in the binomial expansion of (x^3 - 6y)^6 using the binomial theorem. This problem involves combinatorial analysis and algebraic manipulation, suitable for advanced high school students and beyond.

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