If `(1+x+x^(2))^(8)=a_(0)+a_(1)x+a_(2)x^(2)+…a_(16)x^(16)` for all values of x, then `(a_(5))/(100)` is equal to

Answer

To solve the problem, we need to find the coefficient \( a_5 \) in the expansion of \( (1 + x + x^2)^8 \) and then compute \( \frac{a_5}{100} \).

### Step-by-Step Solution

1. **Identify the Expression**:
We start with the expression \( (1 + x + x^2)^8 \).

2. **Use the Multinomial Theorem**:
The multinomial theorem states that:
\[ (x_1 + x_2 + \ldots + x_m)^n = \sum_{k_1 + k_2 + \ldots + k_m = n} \frac{n!}{k_1! k_2! \ldots k_m!} x_1^{k_1} x_2^{k_2} \ldots x_m^{k_m} \]
In our case, \( x_1 = 1 \), \( x_2 = x \), \( x_3 = x^2 \), and \( n = 8 \).

3. **Find the Coefficient of \( x^5 \)**:
We need to find the combinations of \( k_1, k_2, k_3 \) such that:
\[ k_1 + k_2 + k_3 = 8 \quad \text{and} \quad k_2 + 2k_3 = 5 \]
Here, \( k_1 \) is the number of times we choose \( 1 \), \( k_2 \) is the number of times we choose \( x \), and \( k_3 \) is the number of times we choose \( x^2 \).

4. **Express \( k_1 \) in terms of \( k_2 \) and \( k_3 \)**:
From \( k_1 + k_2 + k_3 = 8 \), we can express \( k_1 \) as:
\[ k_1 = 8 - k_2 - k_3 \]
Substitute this into the second equation:
\[ k_2 + 2k_3 = 5 \]

5. **Solve for Non-negative Integer Solutions**:
Rearranging gives:
\[ k_2 = 5 - 2k_3 \]
Now substitute \( k_2 \) back into the equation for \( k_1 \):
\[ k_1 = 8 - (5 - 2k_3) - k_3 = 3 + k_3 \]
Thus, we have:
\[ k_1 = 3 + k_3, \quad k_2 = 5 - 2k_3 \]
We need \( k_1, k_2, k_3 \) to be non-negative integers:
- \( k_3 \geq 0 \)
- \( 5 - 2k_3 \geq 0 \) implies \( k_3 \leq 2.5 \), hence \( k_3 \) can be \( 0, 1, 2 \).

6. **Calculate Valid Combinations**:
- For \( k_3 = 0 \): \( k_1 = 3, k_2 = 5 \)
- For \( k_3 = 1 \): \( k_1 = 4, k_2 = 3 \)
- For \( k_3 = 2 \): \( k_1 = 5, k_2 = 1 \)

7. **Calculate Coefficients**:
Using the multinomial coefficient:
- For \( (3, 5, 0) \):
\[ \frac{8!}{3!5!0!} = \frac{40320}{6 \cdot 120} = 56 \]
- For \( (4, 3, 1) \):
\[ \frac{8!}{4!3!1!} = \frac{40320}{24 \cdot 6 \cdot 1} = 280 \]
- For \( (5, 1, 2) \):
\[ \frac{8!}{5!1!2!} = \frac{40320}{120 \cdot 1 \cdot 2} = 168 \]

8. **Sum the Coefficients**:
Thus, the total coefficient \( a_5 \) is:
\[ a_5 = 56 + 280 + 168 = 504 \]

9. **Calculate \( \frac{a_5}{100} \)**:
Finally, we compute:
\[ \frac{a_5}{100} = \frac{504}{100} = 5.04 \]

### Final Answer:
\[ \frac{a_5}{100} = 5.04 \]