Find the coefficient of
(i) `a^(6) b^(3)` in the expansion of `(2a - (b)/(3) )^9`

Answer

To find the coefficient of \( a^6 b^3 \) in the expansion of \( (2a - \frac{b}{3})^9 \), we can use the Binomial Theorem. The Binomial Theorem states that:

\[ (x + y)^n = \sum_{r=0}^{n} \binom{n}{r} x^{n-r} y^r \]

In our case, we can identify \( x = 2a \) and \( y = -\frac{b}{3} \), and \( n = 9 \).

### Step 1: Write the General Term
The general term \( T_{r+1} \) in the expansion is given by:

\[ T_{r+1} = \binom{n}{r} (2a)^{n-r} \left(-\frac{b}{3}\right)^r \]

Substituting \( n = 9 \):

\[ T_{r+1} = \binom{9}{r} (2a)^{9-r} \left(-\frac{b}{3}\right)^r \]

### Step 2: Simplify the General Term
Now, we can simplify this expression:

\[ T_{r+1} = \binom{9}{r} (2^{9-r} a^{9-r}) \left(-\frac{1}{3}\right)^r b^r \]

This can be rewritten as:

\[ T_{r+1} = \binom{9}{r} 2^{9-r} (-1)^r \frac{b^r}{3^r} a^{9-r} \]

### Step 3: Find the Required Terms
We need to find the term where \( a^{9-r} = a^6 \) and \( b^r = b^3 \). This gives us two equations:

1. \( 9 - r = 6 \) (for \( a \))
2. \( r = 3 \) (for \( b \))

From the first equation, we find:

\[ r = 3 \]

### Step 4: Substitute \( r \) into the General Term
Now we substitute \( r = 3 \) into the general term:

\[ T_{4} = \binom{9}{3} 2^{9-3} (-1)^3 \frac{b^3}{3^3} a^{9-3} \]

This simplifies to:

\[ T_{4} = \binom{9}{3} 2^6 (-1)^3 \frac{b^3}{27} a^6 \]

### Step 5: Calculate the Coefficient
Now we calculate \( \binom{9}{3} \) and \( 2^6 \):

\[ \binom{9}{3} = \frac{9 \times 8 \times 7}{3 \times 2 \times 1} = 84 \]
\[ 2^6 = 64 \]

Now substituting these values back:

\[ T_{4} = 84 \cdot 64 \cdot (-1) \cdot \frac{b^3}{27} a^6 \]

### Step 6: Final Coefficient Calculation
The coefficient of \( a^6 b^3 \) is:

\[ \text{Coefficient} = 84 \cdot 64 \cdot (-1) \cdot \frac{1}{27} \]

Calculating this:

\[ 84 \cdot 64 = 5376 \]
\[ \text{Coefficient} = -\frac{5376}{27} \]

### Step 7: Simplifying the Coefficient
Now, we can simplify \( -\frac{5376}{27} \):

\[ \text{Coefficient} = -199.3333 \text{ (approximately)} \]

However, we can also express it as:

\[ \text{Coefficient} = -\frac{1792}{9} \]

### Final Answer
Thus, the coefficient of \( a^6 b^3 \) in the expansion of \( (2a - \frac{b}{3})^9 \) is:

\[ \boxed{-\frac{1792}{9}} \]