If `(1+x+x^(2))^(8)=a_(0)+a_(1)x+a_(2)x^(2)+.........+a_(16)x^(18)AAx " then "a_(5)`, is equal to……….

Options

A. 502
B. 504
C. 506
D. 508

Explanation

To find the coefficient \( a_5 \) in the expansion of \( (1 + x + x^2)^8 \), we can use the multinomial expansion.

### Step-by-Step Solution:

1. **Understanding the General Term**:
The general term in the expansion of \( (1 + x + x^2)^n \) can be expressed as:
\[ T = \frac{n!}{a! b! c!} (1)^a (x)^b (x^2)^c \]
where \( a + b + c = n \) and \( b + 2c \) gives the power of \( x \) in the term.

2. **Setting Up the Equation**:
For our case, \( n = 8 \), we need to find the coefficient of \( x^5 \). Therefore, we need:
\[ b + 2c = 5 \]
and
\[ a + b + c = 8 \]

3. **Finding Possible Values**:
From the equations:
- From \( a + b + c = 8 \), we can express \( a \) as:
\[ a = 8 - b - c \]
- Substituting this into the first equation gives:
\[ 8 - b - c + b + 2c = 8 \implies c = 5 - b \]

Now we can substitute \( c \) back into the equation \( a + b + c = 8 \):
\[ a + b + (5 - b) = 8 \implies a + 5 = 8 \implies a = 3 \]

Now we can find possible values for \( b \) and \( c \):
- If \( b = 1 \), then \( c = 4 \) (valid)
- If \( b = 3 \), then \( c = 2 \) (valid)
- If \( b = 5 \), then \( c = 0 \) (valid)

Thus, the valid combinations are:
1. \( (a, b, c) = (3, 1, 4) \)
2. \( (a, b, c) = (3, 3, 2) \)
3. \( (a, b, c) = (3, 5, 0) \)

4. **Calculating the Coefficients**:
Now we calculate the coefficients for each valid combination.

- For \( (3, 1, 4) \):
\[ T_1 = \frac{8!}{3! 1! 4!} = \frac{40320}{6 \cdot 1 \cdot 24} = \frac{40320}{144} = 280 \]

- For \( (3, 3, 2) \):
\[ T_2 = \frac{8!}{3! 3! 2!} = \frac{40320}{6 \cdot 6 \cdot 2} = \frac{40320}{72} = 560 \]

- For \( (3, 5, 0) \):
\[ T_3 = \frac{8!}{3! 5! 0!} = \frac{40320}{6 \cdot 120 \cdot 1} = \frac{40320}{720} = 56 \]

5. **Summing the Coefficients**:
Finally, we sum the coefficients:
\[ a_5 = T_1 + T_2 + T_3 = 280 + 560 + 56 = 896 \]

### Final Answer:
Thus, the coefficient \( a_5 \) is equal to \( 896 \).