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)`

Answer

To solve the expression \( \binom{8}{0} - \binom{8}{1} + \binom{8}{2} - \binom{8}{3} + \binom{8}{4} - \binom{8}{5} + \binom{8}{6} - \binom{8}{7} + \binom{8}{8} \), we will calculate each binomial coefficient and then combine them according to the signs given.

### Step-by-step Solution:

1. **Calculate \( \binom{8}{0} \)**:
\[ \binom{8}{0} = 1 \]

2. **Calculate \( \binom{8}{1} \)**:
\[ \binom{8}{1} = 8 \]

3. **Calculate \( \binom{8}{2} \)**:
\[ \binom{8}{2} = \frac{8 \times 7}{2 \times 1} = 28 \]

4. **Calculate \( \binom{8}{3} \)**:
\[ \binom{8}{3} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = 56 \]

5. **Calculate \( \binom{8}{4} \)**:
\[ \binom{8}{4} = \frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1} = 70 \]

6. **Calculate \( \binom{8}{5} \)**:
\[ \binom{8}{5} = \binom{8}{3} = 56 \quad (\text{by symmetry}) \]

7. **Calculate \( \binom{8}{6} \)**:
\[ \binom{8}{6} = \binom{8}{2} = 28 \quad (\text{by symmetry}) \]

8. **Calculate \( \binom{8}{7} \)**:
\[ \binom{8}{7} = \binom{8}{1} = 8 \quad (\text{by symmetry}) \]

9. **Calculate \( \binom{8}{8} \)**:
\[ \binom{8}{8} = 1 \]

10. **Combine all the values according to the signs**:
\[ 1 - 8 + 28 - 56 + 70 - 56 + 28 - 8 + 1 \]

11. **Perform the calculations step-by-step**:
- Start with \( 1 - 8 = -7 \)
- Then, \( -7 + 28 = 21 \)
- Next, \( 21 - 56 = -35 \)
- Then, \( -35 + 70 = 35 \)
- Next, \( 35 - 56 = -21 \)
- Then, \( -21 + 28 = 7 \)
- Next, \( 7 - 8 = -1 \)
- Finally, \( -1 + 1 = 0 \)

Thus, the final value of the expression is:
\[ \boxed{0} \]