Find the coefficient of `x^4` in `(1 + x + x^2)^10`

Answer

To find the coefficient of \( x^4 \) in the expression \( (1 + x + x^2)^{10} \), we can use the multinomial expansion. Here’s a step-by-step solution:

### Step 1: Understanding the Expression
We need to expand \( (1 + x + x^2)^{10} \). The general term in the expansion can be represented using the multinomial theorem.

### Step 2: Multinomial Expansion
The multinomial expansion states that:
\[ (a_1 + a_2 + a_3)^n = \sum_{k_1 + k_2 + k_3 = n} \frac{n!}{k_1! k_2! k_3!} a_1^{k_1} a_2^{k_2} a_3^{k_3} \]
In our case, \( a_1 = 1 \), \( a_2 = x \), and \( a_3 = x^2 \), and \( n = 10 \).

### Step 3: Identify the Terms Contributing to \( x^4 \)
To find the coefficient of \( x^4 \), we need to find combinations of \( k_1 \), \( k_2 \), and \( k_3 \) such that:
\[ k_2 + 2k_3 = 4 \]
and
\[ k_1 + k_2 + k_3 = 10 \]

### Step 4: Solve the System of Equations
From \( k_1 + k_2 + k_3 = 10 \), we can express \( k_1 \) as:
\[ k_1 = 10 - k_2 - k_3 \]
Substituting this into the equation \( k_2 + 2k_3 = 4 \) gives us:
1. \( k_2 + 2k_3 = 4 \)
2. \( k_1 = 10 - k_2 - k_3 \)

We can solve for different values of \( k_3 \):

- **Case 1:** \( k_3 = 0 \)
- \( k_2 + 2(0) = 4 \) → \( k_2 = 4 \)
- \( k_1 = 10 - 4 - 0 = 6 \)
- Contribution: \( \frac{10!}{6!4!0!} = 210 \)

- **Case 2:** \( k_3 = 1 \)
- \( k_2 + 2(1) = 4 \) → \( k_2 = 2 \)
- \( k_1 = 10 - 2 - 1 = 7 \)
- Contribution: \( \frac{10!}{7!2!1!} = 360 \)

- **Case 3:** \( k_3 = 2 \)
- \( k_2 + 2(2) = 4 \) → \( k_2 = 0 \)
- \( k_1 = 10 - 0 - 2 = 8 \)
- Contribution: \( \frac{10!}{8!0!2!} = 45 \)

### Step 5: Sum the Contributions
Now, we sum the contributions 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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