find the value of `""^(8) C_3.`

Options

A. 56
B. `8!`
C. 65
D. `3^8`

Answer

To solve the problem of finding the value of \( \binom{8}{3} \), we will use the formula for combinations. The formula for combinations is given by:

\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]

### Step-by-step Solution:

1. **Identify the values of \( n \) and \( r \)**:
- Here, \( n = 8 \) and \( r = 3 \).

2. **Substitute the values into the combination formula**:
\[ \binom{8}{3} = \frac{8!}{3!(8-3)!} \]
This simplifies to:
\[ \binom{8}{3} = \frac{8!}{3! \cdot 5!} \]

3. **Expand \( 8! \)**:
- We can express \( 8! \) as \( 8 \times 7 \times 6 \times 5! \).
- So, we have:
\[ \binom{8}{3} = \frac{8 \times 7 \times 6 \times 5!}{3! \cdot 5!} \]

4. **Cancel \( 5! \) from the numerator and denominator**:
\[ \binom{8}{3} = \frac{8 \times 7 \times 6}{3!} \]

5. **Calculate \( 3! \)**:
- \( 3! = 3 \times 2 \times 1 = 6 \).

6. **Substitute \( 3! \) back into the equation**:
\[ \binom{8}{3} = \frac{8 \times 7 \times 6}{6} \]

7. **Simplify the expression**:
- The \( 6 \) in the numerator and denominator cancels out:
\[ \binom{8}{3} = 8 \times 7 = 56 \]

### Final Answer:
Thus, the value of \( \binom{8}{3} \) is **56**.