If the coefficient `x^(2) and x^(3)` in the expansion of `(1 + 8x + bx^(2))(1 - 3x)^(9)` in the power of x are equal , then b is :

Options

A. `(54)/(7)`
B. `(27)/(7)`
C. `(108)/(7)`
D. 1

Explanation

To solve the problem, we need to find the value of \( b \) such that the coefficients of \( x^2 \) and \( x^3 \) in the expansion of \( (1 + 8x + bx^2)(1 - 3x)^9 \) are equal.

### Step 1: Expand \( (1 - 3x)^9 \)

Using the binomial theorem, we can expand \( (1 - 3x)^9 \):
\[ (1 - 3x)^9 = \sum_{k=0}^{9} \binom{9}{k} (-3x)^k = \sum_{k=0}^{9} \binom{9}{k} (-3)^k x^k \]

We only need the coefficients for \( x^2 \) and \( x^3 \):
- Coefficient of \( x^2 \) is \( \binom{9}{2} (-3)^2 = 36 \)
- Coefficient of \( x^3 \) is \( \binom{9}{3} (-3)^3 = -84 \)

### Step 2: Find the coefficient of \( x^2 \) in the product

The coefficient of \( x^2 \) in the expansion of \( (1 + 8x + bx^2)(1 - 3x)^9 \) can be calculated by considering the following combinations:
1. \( 1 \cdot \) (coefficient of \( x^2 \) from \( (1 - 3x)^9 \))
2. \( 8x \cdot \) (coefficient of \( x^1 \) from \( (1 - 3x)^9 \))
3. \( bx^2 \cdot \) (coefficient of \( x^0 \) from \( (1 - 3x)^9 \))

Calculating each:
- From \( 1 \): \( 36 \)
- From \( 8x \): \( 8 \cdot \binom{9}{1} (-3) = 8 \cdot (-27) = -216 \)
- From \( bx^2 \): \( b \cdot 1 = b \)

Thus, the coefficient of \( x^2 \) is:
\[ 36 - 216 + b = b - 180 \]

### Step 3: Find the coefficient of \( x^3 \) in the product

The coefficient of \( x^3 \) can be calculated by considering:
1. \( 1 \cdot \) (coefficient of \( x^3 \) from \( (1 - 3x)^9 \))
2. \( 8x \cdot \) (coefficient of \( x^2 \) from \( (1 - 3x)^9 \))
3. \( bx^2 \cdot \) (coefficient of \( x^1 \) from \( (1 - 3x)^9 \))

Calculating each:
- From \( 1 \): \( -84 \)
- From \( 8x \): \( 8 \cdot 36 = 288 \)
- From \( bx^2 \): \( b \cdot (-27) = -27b \)

Thus, the coefficient of \( x^3 \) is:
\[ -84 + 288 - 27b = 204 - 27b \]

### Step 4: Set the coefficients equal

Since the coefficients of \( x^2 \) and \( x^3 \) are equal, we set them equal to each other:
\[ b - 180 = 204 - 27b \]

### Step 5: Solve for \( b \)

Rearranging the equation:
\[ b + 27b = 204 + 180 \]
\[ 28b = 384 \]
\[ b = \frac{384}{28} = \frac{96}{7} \]

### Conclusion

The value of \( b \) is:
\[ b = \frac{96}{7} \]