Statement-1 The number of terms in the expansion of
` (x + (1)/(x) + 1)^(n) " is " (2n +1)`
Statement-2 The number of terms in the expansion of
` (x_(1) + x_(2) + x_(3) + …+ x_(m))^(n) "is " ^(n + m -1)C_(m-1)`.

Answer

To solve the problem, we need to analyze both statements regarding the number of terms in the expansions given.

### Step 1: Analyze Statement 1
We need to find the number of terms in the expansion of \( (x + \frac{1}{x} + 1)^n \).

1. **Rearranging the Expression**:
The expression can be rewritten as:
\[ (x + \frac{1}{x} + 1)^n \]

2. **Identifying the Terms**:
We can treat \( x \), \( \frac{1}{x} \), and \( 1 \) as three distinct terms. The general term in the expansion can be represented as:
\[ T = \frac{n!}{a!b!c!} (x)^a \left(\frac{1}{x}\right)^b (1)^c \]
where \( a + b + c = n \).

3. **Finding the Exponents**:
The exponent of \( x \) in each term will be \( a - b \). The values of \( a \) and \( b \) can vary from \( 0 \) to \( n \). Therefore, \( a - b \) can take values from \( -n \) (when \( a = 0 \) and \( b = n \)) to \( n \) (when \( a = n \) and \( b = 0 \)).

4. **Counting the Distinct Terms**:
The possible values of \( a - b \) are \( -n, -n + 1, \ldots, 0, \ldots, n - 1, n \). This gives us a total of \( 2n + 1 \) distinct values.

Thus, **Statement 1 is correct**: The number of terms in the expansion is \( 2n + 1 \).

### Step 2: Analyze Statement 2
We need to find the number of terms in the expansion of \( (x_1 + x_2 + x_3 + \ldots + x_m)^n \).

1. **Understanding the Formula**:
The formula for the number of terms in the expansion of \( (x_1 + x_2 + \ldots + x_m)^n \) is given by:
\[ \binom{n + m - 1}{m - 1} \]
This formula counts the number of ways to distribute \( n \) indistinguishable objects (the powers) into \( m \) distinguishable boxes (the variables).

2. **Applying the Formula**:
In our case, we have \( m = 3 \) (since we have three variables \( x_1, x_2, x_3 \)). Therefore, substituting \( m \) into the formula:
\[ \text{Number of terms} = \binom{n + 3 - 1}{3 - 1} = \binom{n + 2}{2} \]

Thus, **Statement 2 is also correct**: The number of terms in the expansion is \( \binom{n + m - 1}{m - 1} \).

### Conclusion
Both statements are correct, and Statement 2 provides a correct explanation for Statement 1.

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