find the value of `""^(9) P_3` .

Options

A. 504
B. 309
C. 405
D. 600

Answer

To find the value of \( 9P3 \), we will use the formula for permutations. The formula for permutations is given by:

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

Where:
- \( n \) is the total number of items,
- \( r \) is the number of items to be chosen,
- \( n! \) (n factorial) is the product of all positive integers up to \( n \).

### Step-by-Step Solution:

1. **Identify the values of \( n \) and \( r \)**:
- Here, \( n = 9 \) and \( r = 3 \).

2. **Apply the permutation formula**:
\[ 9P3 = \frac{9!}{(9-3)!} = \frac{9!}{6!} \]

3. **Expand \( 9! \)**:
- We can express \( 9! \) as:
\[ 9! = 9 \times 8 \times 7 \times 6! \]

4. **Substitute back into the formula**:
\[ 9P3 = \frac{9 \times 8 \times 7 \times 6!}{6!} \]

5. **Cancel out \( 6! \)**:
- The \( 6! \) in the numerator and denominator cancels out:
\[ 9P3 = 9 \times 8 \times 7 \]

6. **Calculate the product**:
- First, calculate \( 9 \times 8 = 72 \).
- Then, calculate \( 72 \times 7 = 504 \).

7. **Conclusion**:
- Therefore, the value of \( 9P3 \) is \( 504 \).

### Final Answer:
\[ 9P3 = 504 \]