To find the coefficient of \( x^4 \) in the expansion of \( (1 + x + x^2)^{10} \), we can use the multinomial theorem. The general term in the expansion can be expressed as follows:
### Step 1: Identify the General Term
The general term in the expansion of \( (a + b + c)^n \) is given by:
\[
\frac{n!}{r_1! r_2! r_3!} a^{r_1} b^{r_2} c^{r_3}
\]
where \( r_1 + r_2 + r_3 = n \).
In our case, let \( a = 1 \), \( b = x \), and \( c = x^2 \). Thus, the general term becomes:
\[
\frac{10!}{r_1! r_2! r_3!} (1)^{r_1} (x)^{r_2} (x^2)^{r_3} = \frac{10!}{r_1! r_2! r_3!} x^{r_2 + 2r_3}
\]
### Step 2: Set Up the Equations
We need to find the coefficient of \( x^4 \). Therefore, we need:
\[
r_2 + 2r_3 = 4
\]
Additionally, since the total number of terms must equal 10, we have:
\[
r_1 + r_2 + r_3 = 10
\]
### Step 3: Solve the Equations
From the equations:
1. \( r_1 + r_2 + r_3 = 10 \)
2. \( r_2 + 2r_3 = 4 \)
We can express \( r_1 \) in terms of \( r_2 \) and \( r_3 \):
\[
r_1 = 10 - r_2 - r_3
\]
Now, substituting \( r_2 = 4 - 2r_3 \) into the equation for \( r_1 \):
\[
r_1 = 10 - (4 - 2r_3) - r_3 = 6 + r_3
\]
### Step 4: Find Possible Values for \( r_3 \)
Since \( r_1, r_2, r_3 \) must be non-negative integers, we can analyze the possible values for \( r_3 \):
1. If \( r_3 = 0 \):
- \( r_2 = 4 \)
- \( r_1 = 6 \)
2. If \( r_3 = 1 \):
- \( r_2 = 2 \)
- \( r_1 = 7 \)
3. If \( r_3 = 2 \):
- \( r_2 = 0 \)
- \( r_1 = 8 \)
### Step 5: Calculate the Coefficient for Each Case
Now we calculate the coefficient for each case:
1. **Case 1**: \( (r_1, r_2, r_3) = (6, 4, 0) \)
\[
\text{Coefficient} = \frac{10!}{6!4!0!} = \frac{3628800}{720 \cdot 24} = 210
\]
2. **Case 2**: \( (r_1, r_2, r_3) = (7, 2, 1) \)
\[
\text{Coefficient} = \frac{10!}{7!2!1!} = \frac{3628800}{5040 \cdot 2 \cdot 1} = 360
\]
3. **Case 3**: \( (r_1, r_2, r_3) = (8, 0, 2) \)
\[
\text{Coefficient} = \frac{10!}{8!0!2!} = \frac{3628800}{40320 \cdot 2} = 45
\]
### Step 6: Sum the Coefficients
Finally, we sum the coefficients from all cases:
\[
210 + 360 + 45 = 615
\]
### Final Answer
Thus, the coefficient of \( x^4 \) in \( (1 + x + x^2)^{10} \) is **615**.
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