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In the given example, the problem is to find the number of permutations of a set \( A \), which consists of 4 elements, taken 2 elements at a time.

### Key Concepts:
- **Permutation Formula**: The number of ways to select and arrange \( r \) elements from a set of \( n \) elements is given by:
  \[
  P(n, r) = \frac{n!}{(n - r)!}
  \]
  where \( n! \) denotes the factorial of \( n \), which is the product of all positive integers up to \( n \).

### Given:
- \( n = 4 \) (since the set \( A \) contains 4 elements: {1, 2, 3, 4}),
- \( r = 2 \) (we are choosing 2 elements at a time).

### Step-by-Step Calculation:
1. **Apply the formula**:
   \[
   P(4, 2) = \frac{4!}{(4 - 2)!} = \frac{4!}{2!}
   \]

2. **Evaluate the factorials**:
   - \( 4! = 4 \times 3 \times 2 \times 1 = 24 \)
   - \( 2! = 2 \times 1 = 2 \)

3. **Substitute and divide**:
   \[
   P(4, 2) = \frac{24}{2} = 12
   \]

Thus, the number of permutations is **12**. This means there are 12 ways to arrange 2 elements out of the 4-element set.

Would you like more details or have any questions about this calculation?

Here are 5 related questions to expand your understanding:
1. How would you calculate the permutations if \( r = 3 \) instead of \( 2 \)?
2. What is the difference between permutation and combination?
3. How would you calculate combinations for the same values of \( n \) and \( r \)?
4. What is the significance of the factorial function in permutation and combination formulas?
5. Can you explain how the arrangement changes if we are not allowed to repeat elements?

**Tip**: For larger values of \( n \), using factorials directly can become tedious, so factorial tables or calculators are often used for quicker results.

Find the permutation of the set A taking two elements at a time.

This solution explains how to calculate the permutation of a set of 4 elements, taken 2 at a time, using the formula P(n, r) = n! / (n - r)!. The solution includes detailed steps, including evaluating factorials and dividing them. Suitable for students in Grades 9-12.

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