A delegation of four students is to be selected from a total of 12 students. In how many ways can the delegation be selected.

Answer

To solve the problem of selecting a delegation of 4 students from a total of 12 students, we can use the concept of combinations. The formula for combinations is given by:

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

where \( n \) is the total number of items (students in this case), \( r \) is the number of items to choose (delegates), and \( ! \) denotes factorial.

### Step-by-Step Solution:

1. **Identify the values of \( n \) and \( r \)**:
- Here, \( n = 12 \) (total students) and \( r = 4 \) (students to be selected).

2. **Apply the combination formula**:
- We need to calculate \( C(12, 4) \):
\[ C(12, 4) = \frac{12!}{4!(12-4)!} = \frac{12!}{4! \cdot 8!} \]

3. **Simplify the factorials**:
- We can simplify \( 12! \) as follows:
\[ 12! = 12 \times 11 \times 10 \times 9 \times 8! \]
- Thus, we can rewrite the combination as:
\[ C(12, 4) = \frac{12 \times 11 \times 10 \times 9 \times 8!}{4! \cdot 8!} \]
- The \( 8! \) cancels out:
\[ C(12, 4) = \frac{12 \times 11 \times 10 \times 9}{4!} \]

4. **Calculate \( 4! \)**:
- \( 4! = 4 \times 3 \times 2 \times 1 = 24 \)

5. **Substitute back into the equation**:
\[ C(12, 4) = \frac{12 \times 11 \times 10 \times 9}{24} \]

6. **Perform the multiplication**:
- First, calculate the numerator:
\[ 12 \times 11 = 132 \]
\[ 132 \times 10 = 1320 \]
\[ 1320 \times 9 = 11880 \]
- Now substitute into the equation:
\[ C(12, 4) = \frac{11880}{24} \]

7. **Perform the division**:
\[ C(12, 4) = 495 \]

### Final Answer:
The number of ways to select a delegation of 4 students from a total of 12 students is **495**.