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Factorials are a key concept in combinatorics, often used in the calculation of combinations and permutations. A factorial, noted by the symbol "!", is the product of all positive integers up to a given number. For example, 3 factorial, written as 3!, is calculated as follows:

• 3! = 3 × 2 × 1 = 6

This series of multiplications continues up to the number you are calculating the factorial for.
Factorials simplify the process of counting arrangements and focus particularly on ordering items. In our exercise involving combinations, they serve as a tool to eliminate the effect of different orderings of the same group of items. Understanding factorials is crucial, as they form the backbone of more advanced calculation steps in combinatorics.

The combination formula is used to determine the number of ways to select items from a group, where the order of selection does not matter. This is denoted by the symbol \(C(n, k)\), where \(n\) represents the total number of items to choose from, and \(k\) is how many items you want to choose. The formula for combinations is expressed as:

• \[C(n, k) = \frac{n!}{k!(n-k)!}\]

This formula uses factorials to account for all possible selections and remove duplicates that occur from different item orders.
The exercise provided calculates \(C(3, 2)\), meaning choosing 2 items from a set of 3. Using our formula, we plug in the values like so: \(n = 3\), \(k = 2\):

• \[C(3, 2) = \frac{3!}{2!(3-2)!} = \frac{6}{2 \times 1} = 3\]

This calculation reveals that there are 3 distinct ways to choose 2 items from a total set of 3 without considering the order.

Permutations and combinations are two fundamental concepts in combinatorics, integral in counting arrangements and selections.
**Permutations** address the arrangement of items where the order matters. For example, arranging "ABC" differs from "CAB". The formula for permutations with \(n\) items taken \(k\) at a time is:

• \[P(n, k) = \frac{n!}{(n-k)!}\]

**Combinations** focus on how many ways you can select items where the order doesn't matter, as explained previously with \(C(n, k)\), and focus on "what" you choose.
While permutations evaluate the sequence of arrangement, combinations are non-sequential. Both rely heavily on factorials for calculation, revealing distinct and useful outcomes when dealing with problems involving selections and arrangements.