The value of `""^(40)C_(0) xx ""^(100)C_(40) _ ""^(40)C_(1) xx ""^(99)C_(40) + ""^(40)C_(2) xx ""^(98)C_(40) -"……." + ""^(40)C_(40) xx ""^(60)C_(40)` is equal to `"____"`.

Explanation

To solve the problem, we need to evaluate the expression:

\[ \sum_{k=0}^{40} (-1)^k \binom{40}{k} \binom{100-k}{40} \]

This expression can be interpreted as the coefficient of \(x^{40}\) in the expansion of:

\[ (1+x)^{100} \cdot (1-x)^{40} \]

### Step-by-step Solution:

1. **Understand the Binomial Coefficients**: The expression involves binomial coefficients, which can be interpreted using the binomial theorem. The binomial theorem states that:

\[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]

2. **Set Up the Generating Functions**: We can express the sum as the coefficient of \(x^{40}\) in the expansion of:

\[ (1+x)^{100} \cdot (1-x)^{40} \]

3. **Expand the Terms**: Using the binomial theorem, we can expand both terms:

- For \((1+x)^{100}\):

\[ (1+x)^{100} = \sum_{m=0}^{100} \binom{100}{m} x^m \]

- For \((1-x)^{40}\):

\[ (1-x)^{40} = \sum_{n=0}^{40} \binom{40}{n} (-1)^n x^n \]

4. **Combine the Expansions**: The product of these two expansions gives:

\[ (1+x)^{100} \cdot (1-x)^{40} = \left(\sum_{m=0}^{100} \binom{100}{m} x^m\right) \cdot \left(\sum_{n=0}^{40} \binom{40}{n} (-1)^n x^n\right) \]

5. **Find the Coefficient of \(x^{40}\)**: To find the coefficient of \(x^{40}\), we need to sum the products of coefficients where \(m+n=40\):

\[ \sum_{n=0}^{40} \binom{100}{40-n} \binom{40}{n} (-1)^n \]

6. **Use the Binomial Theorem**: The above sum can be interpreted as:

\[ \text{Coefficient of } x^{40} \text{ in } (1+x)^{100} (1-x)^{40} = \text{Coefficient of } x^{40} \text{ in } (1+x-x)^{100} = \text{Coefficient of } x^{40} \text{ in } (1)^{100} = 1 \]

7. **Conclusion**: Therefore, the value of the original expression is:

\[ \boxed{1} \]