Coefficient of the term independent of x in the expansion of `(1+x)^(2n)(x/(1-x))^(-2n)` is:

Options

A. `""^(2n)C_(n)`
B. `(-1)^(n)""^(2n)C_(n)`
C. `""^(n^(2)-n)C_(n)`
D. 0

Answer

To find the coefficient of the term independent of \( x \) in the expansion of \( (1+x)^{2n} \left( \frac{x}{1-x} \right)^{-2n} \), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ (1+x)^{2n} \left( \frac{x}{1-x} \right)^{-2n} \]
This can be rewritten as:
\[ (1+x)^{2n} \cdot (1-x)^{2n} \cdot x^{2n} \]
This is because \( \left( \frac{x}{1-x} \right)^{-2n} = \left( \frac{1-x}{x} \right)^{2n} = (1-x)^{2n} \cdot x^{-2n} \).

### Step 2: Combine the terms
Now we have:
\[ (1+x)^{2n} \cdot (1-x)^{2n} \cdot x^{2n} \]
We can combine \( (1+x)^{2n} \) and \( (1-x)^{2n} \) using the identity \( (a+b)(a-b) = a^2 - b^2 \):
\[ (1+x)(1-x) = 1 - x^2 \]
Thus, we can express it as:
\[ (1-x^2)^{2n} \cdot x^{2n} \]

### Step 3: Find the term independent of \( x \)
To find the term independent of \( x \) in the expression \( (1-x^2)^{2n} \cdot x^{2n} \), we need to find the coefficient of \( x^{2n} \) in \( (1-x^2)^{2n} \).

### Step 4: Use the Binomial Theorem
Using the Binomial Theorem, the general term in the expansion of \( (1-x^2)^{2n} \) is given by:
\[ \binom{2n}{k} (-1)^k (x^2)^k = \binom{2n}{k} (-1)^k x^{2k} \]
We need the coefficient of \( x^{2n} \), which occurs when \( 2k = 2n \) or \( k = n \).

### Step 5: Calculate the coefficient
Thus, the coefficient of \( x^{2n} \) is:
\[ \binom{2n}{n} (-1)^n \]

### Step 6: Final result
Therefore, the coefficient of the term independent of \( x \) in the original expression is:
\[ \binom{2n}{n} (-1)^n \]