Coefficient of `x^4` in`(x/2-3/(x^2))^(10)` is

Options

A. `(405)/(226)`
B. `(504)/(289)`
C. `(450)/(263)`
D. none of these

Answer

To find the coefficient of \( x^4 \) in the expression \( \left( \frac{x}{2} - \frac{3}{x^2} \right)^{10} \), we will use the Binomial Theorem. Let's break down the solution step by step.

### Step 1: Identify the General Term
The general term \( T_{r+1} \) in the binomial expansion of \( (a + b)^n \) is given by:
\[ T_{r+1} = \binom{n}{r} a^{n-r} b^r \]
In our case, \( n = 10 \), \( a = \frac{x}{2} \), and \( b = -\frac{3}{x^2} \). Thus, the general term becomes:
\[ T_{r+1} = \binom{10}{r} \left( \frac{x}{2} \right)^{10-r} \left( -\frac{3}{x^2} \right)^r \]

### Step 2: Simplify the General Term
Now we simplify the general term:
\[ T_{r+1} = \binom{10}{r} \left( \frac{x^{10-r}}{2^{10-r}} \right) \left( -\frac{3^r}{x^{2r}} \right) \]
This simplifies to:
\[ T_{r+1} = \binom{10}{r} \cdot (-3)^r \cdot \frac{x^{10 - r - 2r}}{2^{10 - r}} = \binom{10}{r} \cdot (-3)^r \cdot \frac{x^{10 - 3r}}{2^{10 - r}} \]

### Step 3: Find the Power of \( x \)
We need the power of \( x \) to be 4:
\[ 10 - 3r = 4 \]
Solving for \( r \):
\[ 10 - 3r = 4 \implies 3r = 10 - 4 \implies 3r = 6 \implies r = 2 \]

### Step 4: Substitute \( r \) into the General Term
Now we substitute \( r = 2 \) back into the general term to find the coefficient:
\[ T_{3} = \binom{10}{2} \cdot (-3)^2 \cdot \frac{x^{10 - 3 \cdot 2}}{2^{10 - 2}} = \binom{10}{2} \cdot 9 \cdot \frac{x^4}{2^8} \]

### Step 5: Calculate the Coefficient
Now we calculate the coefficient:
\[ \binom{10}{2} = \frac{10 \cdot 9}{2 \cdot 1} = 45 \]
Thus, the coefficient is:
\[ \text{Coefficient} = 45 \cdot 9 \cdot \frac{1}{256} = \frac{405}{256} \]

### Final Answer
The coefficient of \( x^4 \) in \( \left( \frac{x}{2} - \frac{3}{x^2} \right)^{10} \) is:
\[ \frac{405}{256} \]