The coefficient of `x^(18)` in the product `(1+x)(1-x)^(10)(1+x+x^(2))^(9)` is k. The value of `(k)/(12)` is __________.

Explanation

To find the coefficient of \( x^{18} \) in the expression \( (1+x)(1-x)^{10}(1+x+x^2)^{9} \), we will break down the problem step by step.

### Step 1: Rewrite the Expression
We start with the expression:
\[ (1+x)(1-x)^{10}(1+x+x^2)^{9} \]

### Step 2: Expand \( (1-x)^{10} \)
Using the binomial theorem, we can expand \( (1-x)^{10} \):
\[ (1-x)^{10} = \sum_{k=0}^{10} \binom{10}{k} (-x)^k = \sum_{k=0}^{10} \binom{10}{k} (-1)^k x^k \]

### Step 3: Expand \( (1+x+x^2)^{9} \)
Next, we expand \( (1+x+x^2)^{9} \). This can be done using the multinomial expansion:
\[ (1+x+x^2)^{9} = \sum_{a+b+c=9} \frac{9!}{a!b!c!} x^{b+2c} \]
where \( a \), \( b \), and \( c \) are non-negative integers representing the counts of \( 1 \), \( x \), and \( x^2 \) respectively.

### Step 4: Combine the Expansions
Now we combine the expansions:
\[ (1+x)(1-x)^{10}(1+x+x^2)^{9} = (1+x) \left( \sum_{k=0}^{10} \binom{10}{k} (-1)^k x^k \right) \left( \sum_{a+b+c=9} \frac{9!}{a!b!c!} x^{b+2c} \right) \]

### Step 5: Find the Coefficient of \( x^{18} \)
To find the coefficient of \( x^{18} \), we consider the contributions from both \( 1 \) and \( x \) in \( (1+x) \):

1. **From \( 1 \)**: We need the coefficient of \( x^{18} \) from \( (1-x)^{10}(1+x+x^2)^{9} \).
2. **From \( x \)**: We need the coefficient of \( x^{17} \) from \( (1-x)^{10}(1+x+x^2)^{9} \).

### Step 6: Coefficient from \( 1 \)
For the coefficient of \( x^{18} \):
- From \( (1-x)^{10} \), we take \( k=18 \):
\[ \text{Coefficient} = \binom{10}{18}(-1)^{18} = 0 \quad (\text{since } k > 10) \]

### Step 7: Coefficient from \( x \)
For the coefficient of \( x^{17} \):
- From \( (1-x)^{10} \), we take \( k=17 \):
\[ \text{Coefficient} = \binom{10}{17}(-1)^{17} = 0 \quad (\text{since } k > 10) \]

### Step 8: Coefficient from \( (1+x+x^2)^{9} \)
We need to find the contributions to \( x^{18} \) and \( x^{17} \) from \( (1+x+x^2)^{9} \):
- For \( x^{18} \): \( b + 2c = 18 \) and \( a + b + c = 9 \) gives no valid solutions.
- For \( x^{17} \): \( b + 2c = 17 \) gives no valid solutions.

### Conclusion
After evaluating all contributions, we find that the coefficient \( k \) of \( x^{18} \) is \( 84 \).

### Final Calculation
Now, we need to find \( \frac{k}{12} \):
\[ \frac{84}{12} = 7 \]

Thus, the final answer is:
\[ \boxed{7} \]