To find the coefficient of \(x^3\) in the expansion of \((1 + x + x^2)^{10}\), we can use the multinomial theorem. The expression \((1 + x + x^2)^{10}\) represents a multinomial expansion.

The general term in the expansion of \((1 + x + x^2)^{10}\) is given by:

\[
T(a, b, c) = \binom{10}{a, b, c} 1^a x^b (x^2)^c
\]

where \(a + b + c = 10\), and \(\binom{10}{a, b, c}\) is the multinomial coefficient:

\[
\binom{10}{a, b, c} = \frac{10!}{a! b! c!}
\]

In the term, \(x^b (x^2)^c = x^{b + 2c}\), and we are interested in the coefficient of the term where \(b + 2c = 3\).

### Step 1: Find all combinations of \(b\) and \(c\) such that \(b + 2c = 3\).

- For \(c = 1\), \(b + 2(1) = 3 \Rightarrow b = 1\).
- For \(c = 0\), \(b + 2(0) = 3 \Rightarrow b = 3\).

Thus, the two valid pairs \((b, c)\) are:

- \((b = 1, c = 1)\)
- \((b = 3, c = 0)\)

### Step 2: Calculate the corresponding values of \(a\), and the multinomial coefficients.

1. For \(b = 1\), \(c = 1\), we have \(a = 10 - b - c = 10 - 1 - 1 = 8\). The multinomial coefficient is:

\[
\binom{10}{8, 1, 1} = \frac{10!}{8!1!1!} = \frac{10 \times 9}{1 \times 1} = 90
\]

2. For \(b = 3\), \(c = 0\), we have \(a = 10 - b - c = 10 - 3 - 0 = 7\). The multinomial coefficient is:

\[
\binom{10}{7, 3, 0} = \frac{10!}{7!3!0!} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120
\]

### Step 3: Add the contributions to the coefficient of \(x^3\).

- From \((b = 1, c = 1)\), the contribution is \(90\).
- From \((b = 3, c = 0)\), the contribution is \(120\).

Thus, the total coefficient of \(x^3\) is:

\[
90 + 120 = 210
\]

### Final Answer:
The coefficient of \(x^3\) in the expansion of \((1 + x + x^2)^{10}\) is **210**.

---

Would you like further details or an explanation of any part of the solution?

Here are 5 related questions to explore further:
1. How do you generalize the multinomial expansion for higher powers?
2. What is the significance of the multinomial coefficient in combinatorics?
3. Can you solve a similar problem with higher powers (e.g., \((1 + x + x^2)^{15}\))?
4. How does the multinomial theorem apply to expansions of terms like \((1 + x + x^2 + x^3)^n\)?
5. What are some applications of the multinomial theorem in probability and statistics?

**Tip:** Always verify the conditions when using the multinomial expansion—especially the sum of the exponents (in this case, \(a + b + c = 10\))!

Discover how to calculate the coefficient of x^3 in the expansion of (1 + x + x^2)^10 using the multinomial theorem. This solution explores combinatorics and algebraic techniques with step-by-step calculations, making it suitable for high school students in Grades 11-12.

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