The value of `((50),(0))((50),(1))+((50),(1))((50),(2))+….+((50),(49))((50),(50))` is :

Options

A. `((100),(51))`
B. `((100),(50))`
C. `((50),(25))`
D. `((50),(25))^(2)`

Answer

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

\[ \sum_{k=0}^{50} \binom{50}{k} \binom{50}{50-k} \]

This expression can be simplified using the properties of binomial coefficients.

### Step-by-Step Solution:

1. **Understanding the Binomial Coefficient**:
The term \(\binom{50}{k}\) represents the number of ways to choose \(k\) items from 50. The term \(\binom{50}{50-k}\) is equal to \(\binom{50}{k}\) due to the symmetry property of binomial coefficients, which states that \(\binom{n}{r} = \binom{n}{n-r}\).

2. **Rewriting the Expression**:
The expression can be rewritten as:
\[ \sum_{k=0}^{50} \binom{50}{k} \binom{50}{k} \]
This is equivalent to:
\[ \sum_{k=0}^{50} \left(\binom{50}{k}\right)^2 \]

3. **Using the Binomial Theorem**:
According to the binomial theorem, the sum of the squares of the binomial coefficients can be expressed as:
\[ \sum_{k=0}^{n} \left(\binom{n}{k}\right)^2 = \binom{2n}{n} \]
For our case, \(n = 50\):
\[ \sum_{k=0}^{50} \left(\binom{50}{k}\right)^2 = \binom{100}{50} \]

4. **Final Calculation**:
Therefore, the value of the original expression is:
\[ \binom{100}{50} \]

### Conclusion:
The final answer to the given expression is:
\[ \binom{100}{50} \]