The total number of different terms in the product `(.^(101)C_(0) - .^(101)C_(1)x+.^(101)C_(2)x^(2)-"….."-.^(101)C_(101)x^(101))(1+x+x^(2)+"…."+x^(100))^(101)` is `"____"`.

Explanation

To find the total number of different terms in the expression

\[ \left( \sum_{k=0}^{101} (-1)^k \binom{101}{k} x^k \right) \left( \sum_{j=0}^{100} x^j \right)^{101}, \]

we can break it down into steps:

### Step 1: Simplify the first part of the expression
The first part of the expression is

\[ \sum_{k=0}^{101} (-1)^k \binom{101}{k} x^k. \]

This is the binomial expansion of \( (1 - x)^{101} \). Therefore, we can rewrite the expression as:

\[ (1 - x)^{101} \cdot \left( \sum_{j=0}^{100} x^j \right)^{101}. \]

### Step 2: Simplify the second part of the expression
The second part, \( \sum_{j=0}^{100} x^j \), is a geometric series and can be expressed as:

\[ \sum_{j=0}^{100} x^j = \frac{1 - x^{101}}{1 - x}. \]

Thus, we can rewrite the entire expression as:

\[ (1 - x)^{101} \cdot \left( \frac{1 - x^{101}}{1 - x} \right)^{101}. \]

### Step 3: Combine the expressions
Now, we can combine the two parts:

\[ (1 - x)^{101} \cdot \frac{(1 - x^{101})^{101}}{(1 - x)^{101}}. \]

The \( (1 - x)^{101} \) terms cancel out, leaving us with:

\[ (1 - x^{101})^{101}. \]

### Step 4: Expand the remaining expression
Now, we need to expand \( (1 - x^{101})^{101} \). The expansion will give us terms of the form \( (-1)^k \binom{101}{k} x^{101k} \) for \( k = 0, 1, 2, \ldots, 101 \).

### Step 5: Identify the powers of \( x \)
The powers of \( x \) in the expansion will be \( 0, 101, 202, \ldots, 10100 \) (which is \( 101 \times 101 \)).

### Step 6: Count the distinct terms
The powers of \( x \) can be represented as \( 101k \) where \( k = 0, 1, 2, \ldots, 101 \). The number of distinct terms corresponds to the values of \( k \):

- For \( k = 0 \): \( 0 \)
- For \( k = 1 \): \( 101 \)
- For \( k = 2 \): \( 202 \)
- ...
- For \( k = 101 \): \( 10100 \)

Thus, the distinct terms are \( 0, 101, 202, \ldots, 10100 \).

### Step 7: Calculate the total number of distinct terms
The total number of distinct terms is given by the number of values \( k \) can take, which is from \( 0 \) to \( 101 \), inclusive. Therefore, the total number of different terms is:

\[ 101 + 1 = 102. \]

### Final Answer
The total number of different terms in the product is \( \boxed{102} \).