The term independent of x in `(1+x)^m (1+1/x)^n` is

Options

A. `""^(m+n)C_(m-1)`
B. `""^(m+n)C_(n)`
C. `""^(m+n)C_(m-n)`
D. None of these

Answer

To find the term independent of \( x \) in the expression \( (1+x)^m (1+\frac{1}{x})^n \), we can follow these steps:

### Step 1: Rewrite the Expression
We start with the expression:
\[ (1+x)^m \cdot \left(1+\frac{1}{x}\right)^n \]
We can rewrite \( \left(1+\frac{1}{x}\right)^n \) as:
\[ \left(\frac{x+1}{x}\right)^n = \frac{(x+1)^n}{x^n} \]
Thus, the expression becomes:
\[ (1+x)^m \cdot \frac{(x+1)^n}{x^n} = \frac{(1+x)^{m+n}}{x^n} \]

### Step 2: General Term of the Expansion
The general term of the expansion of \( (1+x)^{m+n} \) is given by:
\[ T_r = \binom{m+n}{r} x^r \]
where \( r \) is the term number (from 0 to \( m+n \)).

### Step 3: Incorporate the Denominator
Now, we need to incorporate the \( x^{-n} \) from our rewritten expression:
\[ T_r = \binom{m+n}{r} x^r \cdot x^{-n} = \binom{m+n}{r} x^{r-n} \]

### Step 4: Find the Term Independent of \( x \)
For the term to be independent of \( x \), we set the exponent of \( x \) to zero:
\[ r - n = 0 \implies r = n \]

### Step 5: Substitute \( r \) Back into the General Term
Now, we substitute \( r = n \) back into the general term:
\[ T_n = \binom{m+n}{n} x^{n-n} = \binom{m+n}{n} x^0 = \binom{m+n}{n} \]

### Conclusion
Thus, the term independent of \( x \) in the expression \( (1+x)^m (1+\frac{1}{x})^n \) is:
\[ \binom{m+n}{n} \]