the coefficient of `x^(7)` in `(ax - b^(-1) x^(-2))^(11)` is

Options

A. 0
B. 2
C. 3
D. 4

Answer

To find the coefficient of \( x^7 \) in the expansion of \( (ax - b^{-1} x^{-2})^{11} \), we can use the Binomial Theorem. Let's go through the steps:

### Step 1: Identify the Binomial Expansion
The Binomial Theorem states that:
\[ (x + y)^n = \sum_{r=0}^{n} \binom{n}{r} x^{n-r} y^r \]
For our case, we can rewrite the expression as:
\[ (ax - b^{-1} x^{-2})^{11} \]
Here, \( x = ax \) and \( y = -b^{-1} x^{-2} \).

### Step 2: Write the General Term
The general term \( T_{r+1} \) in the expansion can be expressed as:
\[ T_{r+1} = \binom{11}{r} (ax)^{11-r} (-b^{-1} x^{-2})^r \]
This simplifies to:
\[ T_{r+1} = \binom{11}{r} a^{11-r} x^{11-r} (-b^{-1})^r x^{-2r} \]
Combining the powers of \( x \):
\[ T_{r+1} = \binom{11}{r} a^{11-r} (-b^{-1})^r x^{11 - r - 2r} = \binom{11}{r} a^{11-r} (-b^{-1})^r x^{11 - 3r} \]

### Step 3: Set Up the Equation for Coefficient of \( x^7 \)
We want the coefficient of \( x^7 \). Therefore, we set the exponent equal to 7:
\[ 11 - 3r = 7 \]

### Step 4: Solve for \( r \)
Rearranging the equation gives:
\[ 3r = 11 - 7 \implies 3r = 4 \implies r = \frac{4}{3} \]

### Step 5: Analyze the Result
Since \( r \) must be a non-negative integer, \( r = \frac{4}{3} \) is not a valid solution. Therefore, there are no valid \( r \) values that satisfy the condition for the coefficient of \( x^7 \).

### Conclusion
Thus, the coefficient of \( x^7 \) in the expansion of \( (ax - b^{-1} x^{-2})^{11} \) is:
\[ \text{Coefficient} = 0 \]