The value of `""^(10)P_(6)` is

Options

A. 151200
B. 25200
C. 12600
D. 12800

Answer

To solve the problem of finding the value of \( P(10, 6) \), we will use the formula for permutations. The formula for permutations is given by:

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

where \( n \) is the total number of items, and \( r \) is the number of items to choose.

### Step 1: Identify \( n \) and \( r \)
In this case, we have:
- \( n = 10 \)
- \( r = 6 \)

### Step 2: Apply the formula
Using the formula for permutations, we can write:

\[ P(10, 6) = \frac{10!}{(10-6)!} = \frac{10!}{4!} \]

### Step 3: Calculate \( 10! \) and \( 4! \)
Now, we need to calculate \( 10! \) and \( 4! \):
- \( 10! = 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \)
- \( 4! = 4 \times 3 \times 2 \times 1 = 24 \)

### Step 4: Simplify \( P(10, 6) \)
Instead of calculating \( 10! \) fully, we can simplify it by canceling \( 4! \):

\[ P(10, 6) = \frac{10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4!}{4!} \]

The \( 4! \) cancels out:

\[ P(10, 6) = 10 \times 9 \times 8 \times 7 \times 6 \times 5 \]

### Step 5: Calculate the product
Now we will calculate the product step by step:

1. \( 10 \times 9 = 90 \)
2. \( 90 \times 8 = 720 \)
3. \( 720 \times 7 = 5040 \)
4. \( 5040 \times 6 = 30240 \)
5. \( 30240 \times 5 = 151200 \)

Thus,

\[ P(10, 6) = 151200 \]

### Final Answer
The value of \( P(10, 6) \) is \( 151200 \).

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