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The question asks for the coefficient of \(x^2y^4\) in the expansion of \((1 + x + y^2)^7\).

This can be solved by expanding the binomial expression using the multinomial theorem.

### Multinomial Theorem:
The expansion of \((a + b + c)^n\) is given by:
\[
\sum \frac{n!}{k_1!k_2!k_3!} a^{k_1}b^{k_2}c^{k_3}
\]
where \(k_1 + k_2 + k_3 = n\), and \(k_1\), \(k_2\), and \(k_3\) are the powers assigned to each term in the expansion.

### In this case:
We are expanding \((1 + x + y^2)^7\) and we want the coefficient of \(x^2 y^4\). 

1. The \(1\) term has to be raised to \(k_1\).
2. The \(x\) term has to be raised to \(k_2\).
3. The \(y^2\) term has to be raised to \(k_3\).

Since we want \(x^2\) and \(y^4\):
- \(k_2 = 2\) (for \(x^2\))
- \(k_3 = 2\) (since \(y^2\) squared gives \(y^4\), i.e., \(2 \times k_3 = 4\))

Thus, we solve for \(k_1\), using the equation \(k_1 + k_2 + k_3 = 7\):
\[
k_1 + 2 + 2 = 7 \implies k_1 = 3
\]

Now, the coefficient is given by the multinomial coefficient:
\[
\frac{7!}{k_1!k_2!k_3!} = \frac{7!}{3!2!2!}
\]

Let's calculate this expression.The coefficient of \(x^2y^4\) in the expansion of \((1 + x + y^2)^7\) is \(210\).

Would you like more details on the calculation or have any further questions?

Here are some related questions:
1. How does the multinomial theorem differ from the binomial theorem?
2. What is the general form of a multinomial expansion?
3. How can we determine the number of terms in the expansion of \((1 + x + y^2)^n\)?
4. How would the expansion change if the exponent were higher than 7?
5. What if instead of \(y^2\), the term was \(y^3\)? How would that affect the calculation?

**Tip:** The multinomial theorem is a powerful tool for expanding expressions with more than two terms, and it simplifies solving problems involving combinations of variables like this one.

Find the coefficient of x^2y^4 in the expansion of (1 + x + y^2)^7

This page provides a detailed solution to finding the coefficient of x^2y^4 in the expansion of (1 + x + y^2)^7 using the multinomial theorem. Learn the step-by-step process and calculations suitable for advanced high school students (Grades 11-12).

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