91Ó°ÊÓ

Combinatorics is the branch of mathematics that deals with counting, arrangement, and combination of objects. In this exercise, we use combinatorial methods to determine all possible pairings of job applicants. To think about this systematically, consider each applicant as a unique object. We want to calculate how many different pairs can be formed from these five applicants: three men and two women.

In basic combinatorics, when we are selecting two objects from five, we use combinations since the order in which we select them doesn't matter here. The number of possible combinations of two objects from a set of five is given by the formula: \[ \binom{5}{2} = \frac{5 \times 4}{2 \times 1} = 10. \]

This means there are 10 unique ways to select two applicants out of five, which matches the list of pairs provided in the sample space. This process of listing pairs is an example of using combinatorics to solve real-world problems.

The concept of a sample space is foundational in probability theory. It refers to the set of all possible outcomes of a particular experiment or process. In our context, the experiment is selecting two applicants from five.

Here, the sample space \( S \) includes every possible pair of applicants that can be chosen. Each element of \( S \) represents one possible outcome of the job selection process.
For this problem:

• We listed all possible pairs, such as \( \{ M_1, M_2 \} \), \( \{ M_1, M_3 \} \), up to \( \{ W_1, W_2 \} \).
• The comprehensive list is: \( S = \{ \{ M_1, M_2 \}, \{ M_1, M_3 \}, \{ M_1, W_1 \}, \{ M_1, W_2 \}, \{ M_2, M_3 \}, \{ M_2, W_1 \}, \{ M_2, W_2 \}, \{ M_3, W_1 \}, \{ M_3, W_2 \}, \{ W_1, W_2 \} \} \).

Understanding the sample space helps in identifying any subset within it and is crucial for calculating probabilities.

Set operations in mathematics involve combining, intersecting, or differentiating collections of elements, which helps in organizing and analyzing data.

In this exercise, we explore several set operations:

• **Union (\(A \cup B\))** - This combines the elements from two sets. For example, \( A \cup B \) includes outcomes from either \( A \) or \( B \), or both. Here it equates to the full sample space since it covers all combinations of selections.
• **Intersection (\(A \cap B\))** - This contains only elements that are present in both sets. For instance, \( A \cap B \), which seeks selections that are simultaneously in \( A \) and \( B \), results in an empty set \( \emptyset \) because it is not possible to select two men and at least one woman at the same time.
• **Complement (\(\bar{B}\))** - This includes everything not in a set. \( \bar{B} \) symbolizes outcomes without women, thus primarily focuses on men, aligning \( \bar{B} \) to \( A \) in this example since they both only contain men.
• **Other Combinations - (\(A \cap \bar{B}\))** - Such combinations explore more specific criteria like selecting two men, naturally overlapping \( A \) and \( \bar{B} \), generating an identical set as that of two men.

Using set operations allows us to categorize and interpret the different selection scenarios presented.