Twelve students complete for a race. The number of ways in which first three prizes can be taken is

Options

A. `(12)!-3`
B. `(12)!3!`
C. `3!`
D. `12.11.10`

Answer

To find the number of ways in which the first three prizes can be awarded to twelve students, we will use the concept of permutations.

### Step-by-Step Solution:

1. **Identify the total number of students (n)**:
We have 12 students competing for the prizes.
\[ n = 12 \]

2. **Identify the number of prizes (r)**:
We are interested in the first three prizes.
\[ r = 3 \]

3. **Use the permutation formula**:
The number of ways to arrange \( r \) objects from \( n \) objects is given by the permutation formula:
\[ P(n, r) = \frac{n!}{(n - r)!} \]
In our case, we need to calculate \( P(12, 3) \).

4. **Calculate \( P(12, 3) \)**:
\[ P(12, 3) = \frac{12!}{(12 - 3)!} = \frac{12!}{9!} \]
This simplifies to:
\[ P(12, 3) = 12 \times 11 \times 10 \]

5. **Perform the multiplication**:
\[ 12 \times 11 = 132 \]
\[ 132 \times 10 = 1320 \]

6. **Final answer**:
Therefore, the number of ways in which the first three prizes can be awarded is:
\[ \boxed{1320} \]