What is the value of `""^(8)C_(0)-""^(8)C_(1)+""^(8)C_(2)-""^(8)C_(3)+""^(8)C_(4)-""^(8)C_(5)+""^(8)C_(6)-""^(8)C_(7)+""^(8)C_(8)`

Options

A. 0
B. 1
C. 2
D. `2^(8)`

Explanation

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

\[ {8 \choose 0} - {8 \choose 1} + {8 \choose 2} - {8 \choose 3} + {8 \choose 4} - {8 \choose 5} + {8 \choose 6} - {8 \choose 7} + {8 \choose 8} \]

### Step 1: Recognize the Pattern
This expression can be recognized as the expansion of \((1 - 1)^8\) using the Binomial Theorem. According to the Binomial Theorem, we have:

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

### Step 2: Apply the Binomial Theorem
In our case, let \(a = 1\) and \(b = -1\). Therefore, we can express the left-hand side as:

\[ (1 - 1)^8 = \sum_{k=0}^{8} {8 \choose k} (1)^{8-k} (-1)^k \]

### Step 3: Simplify the Expression
This simplifies to:

\[ (1 - 1)^8 = 0^8 = 0 \]

### Step 4: Conclusion
Thus, the value of the original expression is:

\[ {8 \choose 0} - {8 \choose 1} + {8 \choose 2} - {8 \choose 3} + {8 \choose 4} - {8 \choose 5} + {8 \choose 6} - {8 \choose 7} + {8 \choose 8} = 0 \]

### Final Answer
The value of the expression is \(0\).

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