The sum of the series
`.^(20)C_(0)-.^(20)C_(1)+ .^(20)C_(2)-.^(20)C_(3)+...-.+ .^(20)C_(10)` is -

Options

A. `-.^(20)C_(10)`
B. `1/2 .^(20)C_(10)`
C. 0
D. `.^(20)C_(10)`

Answer

To solve the problem of finding the sum of the series \( \binom{20}{0} - \binom{20}{1} + \binom{20}{2} - \binom{20}{3} + \ldots - \binom{20}{10} \), we can follow these steps:

### Step 1: Recognize the Pattern
The series can be represented as:
\[ S = \sum_{r=0}^{10} (-1)^r \binom{20}{r} \]
This indicates that we are summing the binomial coefficients with alternating signs.

### Step 2: Use Binomial Theorem
Recall the binomial theorem:
\[ (1 + x)^n = \sum_{r=0}^{n} \binom{n}{r} x^r \]
We can use this theorem to evaluate our series by substituting \( x = -1 \):
\[ (1 - 1)^{20} = \sum_{r=0}^{20} \binom{20}{r} (-1)^r \]
This simplifies to:
\[ 0 = \sum_{r=0}^{20} \binom{20}{r} (-1)^r \]

### Step 3: Split the Sum
We can split the sum into two parts:
\[ 0 = \left( \sum_{r=0}^{10} \binom{20}{r} (-1)^r \right) + \left( \sum_{r=11}^{20} \binom{20}{r} (-1)^r \right) \]
Let \( S = \sum_{r=0}^{10} \binom{20}{r} (-1)^r \). The second part can be rewritten using the property of binomial coefficients:
\[ \sum_{r=11}^{20} \binom{20}{r} (-1)^r = -\sum_{r=0}^{9} \binom{20}{20-r} (-1)^{20-r} = -\sum_{r=0}^{9} \binom{20}{r} (-1)^{r} = -S \]

### Step 4: Set Up the Equation
Now we have:
\[ 0 = S - S \implies 0 = 2S \]
This implies:
\[ S = 0 \]

### Step 5: Find the Relevant Sum
However, we need to find the sum only up to \( \binom{20}{10} \). Since the series from \( \binom{20}{0} \) to \( \binom{20}{20} \) sums to zero, we can conclude that:
\[ S = \binom{20}{0} - \binom{20}{1} + \binom{20}{2} - \binom{20}{3} + \ldots - \binom{20}{10} = -\binom{20}{11} - \binom{20}{12} - \ldots - \binom{20}{20} \]
Thus, the sum we are looking for is half of the total sum of the coefficients from \( \binom{20}{0} \) to \( \binom{20}{10} \).

### Step 6: Final Calculation
The final result, therefore, is:
\[ S = -\frac{1}{2} \binom{20}{10} \]

### Conclusion
The sum of the series \( \binom{20}{0} - \binom{20}{1} + \binom{20}{2} - \binom{20}{3} + \ldots - \binom{20}{10} \) is \( -\frac{1}{2} \binom{20}{10} \).