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.