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Factorial notation is a mathematical convention for expressing the product of a series of descending natural numbers. To represent a number's factorial, we use the exclamation mark symbol \textbf{(!)}. For example, the factorial of 5, written as \(5!\), is calculated by multiplying 5 by every positive integer less than itself: \[5! = 5 \times 4 \times 3 \times 2 \times 1 = 120.\]

Factorials are exceedingly important in combinatorics because they are used in formulas to count the number of ways things can be arranged or combined. They provide a simple way to calculate the permutations or combinations of a given set, a fundamental concept in probability and statistics as well. A common application in combinatorial problems is when we have to find the number of possible ways to arrange or select items without repetition.

Understanding this mathematical concept is vital when solving problems involving combinations, as problem step 1 demonstrated, where factorial notation was used to calculate the number of subsets of a set.

Combinations are selections made by taking some or all objects without considering the order of the objects. This is a key concept in combinatorics, which is a branch of mathematics focused on counting, arrangement, and grouping methods. To calculate combinations, the formula \(_nC_r = \frac{n!}{r!(n-r)!}\) is used, where \(n\) is the total number of items to choose from, \(r\) is the number of items to be chosen, and \(n!\) is the factorial of \(n\).

The distinction between combinations and permutations lies in the importance of order; combinations do not account for different orders, whereas permutations do. For example, in the textbook exercise, the number of ways to select 5 elements from a set of 30, without regard to the order, is expressed as \(_{30}C_{5}\). This approach to counting is essential when solving problems that ask for the number of ways to choose items from a larger set when the order doesn't matter—as seen in the provided steps for calculating subsets.

Set theory is a fundamental theory in mathematics that deals with the collection of objects or elements known as sets. This concept is the basis for all of mathematics and is particularly useful in combinatorics. In set theory, the size or cardinality of a set is denoted as \(|S|\), and it refers to the number of elements in set \(S\).

Within this framework, the notion of a subset is critical. A subset is any set where all elements are contained within a larger set. The power set of any given set \(S\) is the set of all possible subsets of \(S\), including both the empty set and \(S\) itself. When working with subsets, we often want to determine the number of specific subsets that meet certain criteria, such as having a fixed number of elements or a particular smallest element.

In the given exercise solutions, understanding set theory allowed the calculation of subsets of a particular set that satisfy various conditions like having a fixed cardinality of 5 or having a smallest element less than or equal to 5. These applications of set theory simplify the process of addressing complex combinatorial problems.