The term independent ofx in the expansion of `(1+x)^10*(1+1/x)^10` is

Options

A. `""^(22)C_(10)`
B. 0
C. `""^(22)C_(11)`
D. none of these

Answer

To find the term independent of \( x \) in the expansion of \( (1+x)^{10} \cdot \left(1+\frac{1}{x}\right)^{10} \), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ (1+x)^{10} \cdot \left(1+\frac{1}{x}\right)^{10} \]
We can rewrite \( \left(1+\frac{1}{x}\right)^{10} \) as \( \left(\frac{x+1}{x}\right)^{10} \):
\[ (1+x)^{10} \cdot \left(\frac{x+1}{x}\right)^{10} = (1+x)^{10} \cdot \frac{(x+1)^{10}}{x^{10}} = \frac{(1+x)^{20}}{x^{10}} \]

### Step 2: Expand the numerator using the Binomial Theorem
Using the Binomial Theorem, we can expand \( (1+x)^{20} \):
\[ (1+x)^{20} = \sum_{k=0}^{20} \binom{20}{k} x^k \]

### Step 3: Combine the expansions
Now, substituting back into our expression:
\[ \frac{(1+x)^{20}}{x^{10}} = \sum_{k=0}^{20} \binom{20}{k} x^{k-10} \]

### Step 4: Find the term independent of \( x \)
To find the term independent of \( x \), we need \( k - 10 = 0 \), which gives us \( k = 10 \).

### Step 5: Calculate the coefficient
The coefficient of the term where \( k = 10 \) is:
\[ \binom{20}{10} \]

### Conclusion
Thus, the term independent of \( x \) in the expansion is:
\[ \binom{20}{10} \]