What is the coefficient of `x^(17)` in the expansion of `(3x-(x^(3))/(6))^(9)` ?

Options

A. `(189)/(8)`
B. `(567)/(2)`
C. `(21)/(16)`
D. None of these

Explanation

To find the coefficient of \( x^{17} \) in the expansion of \( \left( 3x - \frac{x^3}{6} \right)^9 \), we can use the Binomial Theorem.

### Step-by-Step Solution:

1. **Identify the General Term**:
The general term \( T_{r+1} \) in the expansion of \( (a + b)^n \) is given by:
\[ T_{r+1} = \binom{n}{r} a^{n-r} b^r \]
In our case, \( a = 3x \) and \( b = -\frac{x^3}{6} \), and \( n = 9 \).

2. **Write the General Term for Our Expression**:
Thus, the general term becomes:
\[ T_{r+1} = \binom{9}{r} (3x)^{9-r} \left(-\frac{x^3}{6}\right)^r \]

3. **Simplify the General Term**:
Expanding this, we have:
\[ T_{r+1} = \binom{9}{r} (3^{9-r} x^{9-r}) \left(-\frac{1}{6^r} x^{3r}\right) \]
This simplifies to:
\[ T_{r+1} = \binom{9}{r} 3^{9-r} \left(-\frac{1}{6^r}\right) x^{(9-r) + 3r} \]
\[ = \binom{9}{r} 3^{9-r} \left(-\frac{1}{6^r}\right) x^{9 + 2r} \]

4. **Set the Power of x to 17**:
We need the power of \( x \) to equal 17:
\[ 9 + 2r = 17 \]
Solving for \( r \):
\[ 2r = 17 - 9 = 8 \quad \Rightarrow \quad r = 4 \]

5. **Substitute r back into the General Term**:
Now substitute \( r = 4 \) into the general term:
\[ T_{5} = \binom{9}{4} 3^{9-4} \left(-\frac{1}{6^4}\right) x^{17} \]
\[ = \binom{9}{4} 3^{5} \left(-\frac{1}{6^4}\right) x^{17} \]

6. **Calculate the Coefficient**:
Now we need to calculate the coefficient:
\[ \text{Coefficient} = \binom{9}{4} \cdot 3^5 \cdot \left(-\frac{1}{6^4}\right) \]

- Calculate \( \binom{9}{4} \):
\[ \binom{9}{4} = \frac{9!}{4!(9-4)!} = \frac{9 \times 8 \times 7 \times 6}{4 \times 3 \times 2 \times 1} = 126 \]
- Calculate \( 3^5 \):
\[ 3^5 = 243 \]
- Calculate \( 6^4 \):
\[ 6^4 = 1296 \]

Now, substituting these values back:
\[ \text{Coefficient} = 126 \cdot 243 \cdot \left(-\frac{1}{1296}\right) \]

7. **Final Calculation**:
\[ = -\frac{126 \cdot 243}{1296} \]
Simplifying:
\[ = -\frac{126 \cdot 243}{6^4} = -\frac{126 \cdot 243}{1296} = -\frac{126 \cdot 243}{6 \cdot 6 \cdot 6 \cdot 6} \]
After performing the multiplication and division:
\[ = -\frac{30738}{1296} = -\frac{189}{8} \]

Thus, the coefficient of \( x^{17} \) in the expansion is:
\[ \boxed{-\frac{189}{8}} \]