STATEMENT - 1 : The term independent of x in the expansion of `(x+1/x+2)^(m)` is `((2m) !)/((m !)^(2))`
STATEMENT - 2 : The coefficient of `x^(b)` in the expansion of `(1+x)^(n)` is `.^(n)C_(b)`.

Options

A. STATEMENT - 1 is true, STATEMENT - 2 is true and STATEMENT - 2 is correct explanation for STATEMENT - 1.
B. STATEMENT - 1 is true, STATEMENT - 2 is true and STATEMENT - 2 is not correct explanation for STATEMENT - 1.
C. STATEMENT-1 is true, STATEMENT-2 is false
D. STATEMENT-1 is false, STATEMENT-2 is true

Answer

To solve the given problem, we need to analyze both statements one by one and verify their correctness.

### Statement 1:
The term independent of \( x \) in the expansion of \( (x + \frac{1}{x} + 2)^m \) is \( \frac{(2m)!}{(m!)^2} \).

#### Step 1: Rewrite the expression
We start with the expression:
\[ (x + \frac{1}{x} + 2)^m \]
This can be rewritten as:
\[ (x + \frac{1}{x})^m + 2^m \]

#### Step 2: Use the Binomial Theorem
Using the Binomial Theorem, we can expand \( (x + \frac{1}{x})^m \):
\[ (x + \frac{1}{x})^m = \sum_{k=0}^{m} \binom{m}{k} x^{m-k} \left(\frac{1}{x}\right)^k = \sum_{k=0}^{m} \binom{m}{k} x^{m-2k} \]

#### Step 3: Find the term independent of \( x \)
The term independent of \( x \) occurs when the exponent of \( x \) is zero:
\[ m - 2k = 0 \implies k = \frac{m}{2} \]
This means \( m \) must be even for \( k \) to be an integer. Let \( m = 2n \) for some integer \( n \).

#### Step 4: Calculate the coefficient
The coefficient of the term \( k = n \) is:
\[ \binom{2n}{n} \]
Thus, the term independent of \( x \) is:
\[ \binom{2n}{n} = \frac{(2n)!}{(n!)^2} \]
Substituting back \( n = \frac{m}{2} \):
\[ \frac{(2m)!}{(m!)^2} \]
This confirms that Statement 1 is **true**.

### Statement 2:
The coefficient of \( x^b \) in the expansion of \( (1 + x)^n \) is \( \binom{n}{b} \).

#### Step 1: Use the Binomial Theorem
The Binomial Theorem states:
\[ (1 + x)^n = \sum_{k=0}^{n} \binom{n}{k} x^k \]

#### Step 2: Identify the coefficient of \( x^b \)
From the expansion, the coefficient of \( x^b \) is clearly:
\[ \binom{n}{b} \]
This confirms that Statement 2 is also **true**.

### Conclusion
Both statements are true. Therefore, we can conclude that:

- **Statement 1 is true.**
- **Statement 2 is true and provides a correct explanation for Statement 1.**