The value of `""^(10)P_(3)` is

Options

A. 504
B. 308
C. 720
D. 600

Answer

To find the value of \( ^{10}P_3 \), we can use the formula for permutations, which is given by:

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

Here, \( n = 10 \) and \( r = 3 \).

### Step 1: Substitute the values into the formula
We substitute \( n \) and \( r \) into the formula:

\[ ^{10}P_3 = \frac{10!}{(10-3)!} = \frac{10!}{7!} \]

### Step 2: Expand \( 10! \)
Next, we can expand \( 10! \):

\[ 10! = 10 \times 9 \times 8 \times 7! \]

### Step 3: Substitute back into the equation
Now we substitute this back into our permutation formula:

\[ ^{10}P_3 = \frac{10 \times 9 \times 8 \times 7!}{7!} \]

### Step 4: Cancel \( 7! \)
The \( 7! \) in the numerator and denominator cancels out:

\[ ^{10}P_3 = 10 \times 9 \times 8 \]

### Step 5: Calculate the product
Now, we calculate the product:

\[ 10 \times 9 = 90 \]
\[ 90 \times 8 = 720 \]

### Final Answer
Thus, the value of \( ^{10}P_3 \) is:

\[ ^{10}P_3 = 720 \]