Find the value of `9P_3`

Options

A. 504
B. 239
C. 405
D. 600

Answer

To find the value of \( 9P_3 \), we will use the formula for permutations, which is given by:

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

### Step 1: Identify the values of \( n \) and \( r \)
In this case, \( n = 9 \) and \( r = 3 \).

### Step 2: Substitute the values into the formula
Using the formula for permutations, we substitute \( n \) and \( r \):

\[ 9P_3 = \frac{9!}{(9 - 3)!} \]

### Step 3: Simplify the expression
Calculate \( 9 - 3 \):

\[ 9 - 3 = 6 \]

Now, substitute this back into the equation:

\[ 9P_3 = \frac{9!}{6!} \]

### Step 4: Expand \( 9! \)
The factorial \( 9! \) can be expressed as:

\[ 9! = 9 \times 8 \times 7 \times 6! \]

### Step 5: Substitute back into the equation
Now substitute \( 9! \) back into the equation:

\[ 9P_3 = \frac{9 \times 8 \times 7 \times 6!}{6!} \]

### Step 6: Cancel out \( 6! \)
The \( 6! \) in the numerator and denominator cancels out:

\[ 9P_3 = 9 \times 8 \times 7 \]

### Step 7: Calculate the product
Now we calculate \( 9 \times 8 \times 7 \):

\[ 9 \times 8 = 72 \]
\[ 72 \times 7 = 504 \]

Thus, the value of \( 9P_3 \) is:

\[ 9P_3 = 504 \]

### Final Answer
The value of \( 9P_3 \) is \( 504 \).

---