a. To determine if there is a majority winner, we need to check if any candidate has more than half of the total votes. In this case, the total number of votes is 2 + 5 + 4 + 2 = 13. Since no candidate has more than half of the total votes (which would be 6.5 votes), there is no majority winner.
To find the plurality winner, we need to identify the candidate with the highest number of votes. In this case, candidate BAC has the most votes with 5 votes. Therefore, the plurality winner is candidate BAC.
b. The Borda count method assigns points to each candidate based on their rank in each vote. The candidate with the highest total points wins. Assigning points based on the ranks in each vote:
- ACB: A = 0, C = 2, B = 1
- BAC: B = 0, A = 2, C = 1
- CBA: C = 0, B = 2, A = 1
- CAB: C = 0, A = 2, B = 1
Calculating the total points for each candidate:
- A: 0 + 2 + 1 + 2 = 5
- B: 1 + 0 + 2 + 1 = 4
- C: 2 + 1 + 0 + 0 = 3
Based on the Borda count method, candidate A has the highest total points and wins the election.
c. To determine the winner using the elimination method, we first eliminate the candidate with the most last-place votes. In this case, candidate C has the most last-place votes with 4 votes. Therefore, candidate C is eliminated.
After eliminating candidate C, we have a runoff between candidates A and B:
- ACB: A = 0, C = X (eliminated), B = 1
- BAC: B = 0, A = X (eliminated), C = 1
- CAB: C = X (eliminated), A = 2, B = 1
Now we count the votes for the runoff:
- ACB: B = 1
- BAC: C = 1
- CAB: A = 2, B = 1
Candidate A has the most votes in the runoff and wins the election.
Therefore, using the elimination method, candidate A wins the election.