`""^(12)P_(3) = ?`

Options

A. 210
B. 455
C. 2730
D. None of these

Answer

To solve the problem \(^{12}P_{3}\), we will use the formula for permutations, which is given by:

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

In this case, \(n = 12\) and \(r = 3\).

### Step 1: Apply the Permutation Formula
Using the formula, we can write:

\[ ^{12}P_{3} = \frac{12!}{(12 - 3)!} = \frac{12!}{9!} \]

### Step 2: Simplify the Factorials
Next, we can simplify \(12!\) in terms of \(9!\):

\[ 12! = 12 \times 11 \times 10 \times 9! \]

Now substituting this back into our equation gives:

\[ ^{12}P_{3} = \frac{12 \times 11 \times 10 \times 9!}{9!} \]

### Step 3: Cancel the \(9!\)
The \(9!\) in the numerator and denominator cancels out:

\[ ^{12}P_{3} = 12 \times 11 \times 10 \]

### Step 4: Calculate the Product
Now, we need to calculate \(12 \times 11 \times 10\):

1. First, calculate \(11 \times 10\):

\[ 11 \times 10 = 110 \]

2. Now multiply this result by 12:

\[ 12 \times 110 = 1320 \]

### Final Answer
Thus, the value of \(^{12}P_{3}\) is:

\[ ^{12}P_{3} = 1320 \]