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Preference order is a fundamental concept in voting theory. It describes how voters rank candidates based on their preferences.
In the given exercise, each member of the Wohascum County Board of Commissioners ranked the candidates A, B, and C in a specific order from most to least preferred.
This results in various combinations of ordered preferences.
When analyzing preference orders, it's important to collect all the rankings to understand the overall preference of the voters. In our case, the important preference orders led to the following system of equations during the exercise:
- 11 members preferred A over B.
- 12 members preferred C over A.
- 14 members preferred B over C.

These preferences are decoded into equations that then help in finding how each candidate stands among the voters based on their position in the list.

Forming an equation system helps solve many problems involving preferences.
In this exercise, three key equations were set up to reflect the given preferences:

- Let: - x be the number of members who prefer the order A > B > C.
- y be the number of members who prefer the order B > C > A.
- z be the number of members who prefer the order C > A > B.

The following equations are derived from the preferences:
1. x + (14 - z) + z = 20 (All voters are accounted for)
2. x + z = 11 (Preference A over B)
3. z + y = 12 (Preference C over A)
4. y + (11 - x) = 14 (Preference B over C)

The goal is to solve these equations to determine the exact numbers of voters that preferred each order.

Candidate elimination refers to the step in voting where one candidate is removed from consideration to simplify the decision-making process.
In the exercise, it was suggested that candidate B should withdraw, enabling a runoff between A and C.
However, B protested, which led to further analysis, revealing that most members preferred B over C.

By doing this additional analysis with candidate elimination, we see why simply removing a candidate based on partial data can lead to a misrepresentation of the voter's true preferences.
This step highlights the importance of comprehensive data and detailed analysis in determining candidate viability.

Ballot analysis involves looking deeply into voting data to find patterns and preferences.
In the exercise, the ballots were analyzed to determine the exact number of members supporting particular orders of preference.
Each possible preference was collected and compared:
- The number of members who preferred A over B (11 members)
- The number of members who preferred C over A (12 members)
- The number of members who preferred B over C (14 members)

A step-wise solution involves forming equations and analyzing the votes.
This includes substitution and solving systems of equations. Consequently, the analysis revealed:
- x (A > B > C) = 2 members
- y (B > C > A) = 3 members
- z (C > A > B) = 9 members.
The thorough ballot analysis allowed for a deeper understanding of the preferences and the final distribution of first-choice votes.