The value of `sum_(r=0)^(2n)(-1)^(r)*(""^(2n)C_(r))^(2)` is equal to :

Options

A. `""^(4n)C_(2n)`
B. `""^(2n)C_(n)`
C. `(-1)^(n)*""^(2n)C_(n)`
D. None of these

Answer

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

\[ S = \sum_{r=0}^{2n} (-1)^r \binom{2n}{r}^2 \]

### Step 1: Understanding the Binomial Coefficient
The binomial coefficient \(\binom{n}{r}\) represents the number of ways to choose \(r\) elements from \(n\) elements. In our case, we are squaring it, which means we are considering the combinations in pairs.

### Step 2: Using the Binomial Theorem
We can utilize the binomial theorem to express the sum. According to the binomial theorem:

\[ (1 + x)^{2n} = \sum_{r=0}^{2n} \binom{2n}{r} x^r \]

We can also consider the expansion of \((1 - x)^{2n}\):

\[ (1 - x)^{2n} = \sum_{r=0}^{2n} \binom{2n}{r} (-x)^r = \sum_{r=0}^{2n} (-1)^r \binom{2n}{r} x^r \]

### Step 3: Forming the Product
Now, we can multiply the two expansions:

\[ (1 + x)^{2n} (1 - x)^{2n} = ((1 + x)(1 - x))^{2n} = (1 - x^2)^{2n} \]

### Step 4: Coefficient Extraction
The coefficient of \(x^{2r}\) in \((1 - x^2)^{2n}\) can be expressed as:

\[ \sum_{r=0}^{2n} \binom{2n}{r}^2 (-1)^r \]

This means we need to find the coefficient of \(x^{0}\) in \((1 - x^2)^{2n}\), which is simply \(\binom{2n}{n}\).

### Step 5: Final Calculation
Thus, we have:

\[ S = \sum_{r=0}^{2n} (-1)^r \binom{2n}{r}^2 = \binom{2n}{n} \]

### Conclusion
The value of the sum is:

\[ \sum_{r=0}^{2n} (-1)^r \binom{2n}{r}^2 = \binom{2n}{n} \]