Question
Candidates Joy, Ned, and Harvey are on a ballot to be the county treasurer.The table shows preference orders for six people’s votes (J = Joy, N = Ned, H = Harvey). How many unique preference orders are represented in the table? Voter 1 Voter 2 Voter 3 Voter 4 Voter 5 Voter 61st J N J J H N2nd N J N N N J3rd H H H H J H(1 point)Responsesthreethreefourfourfivefivesix
Answers
GPT-4o mini
To determine the number of unique preference orders represented in the table, we can analyze the preference orders given by each voter, which are listed as follows:
- Voter 1: J > N > H
- Voter 2: N > J > H
- Voter 3: J > N > H
- Voter 4: J > H > N
- Voter 5: N > N > J
- Voter 6: H > H > H
Consolidating this information, we can identify the unique preference orders:
1. J > N > H (Voters 1 and 3)
2. N > J > H (Voter 2)
3. J > H > N (Voter 4)
4. N > J > H (Voter 5) - this is actually the same as preference order 2, so we don't count it again.
5. H > H > H (Voter 6) - this appears to be an invalid order because it doesn't rank all candidates, but we can treat it as a preference if needed.
Since J > N > H and N > J > H are counted only once, we have:
- From Voter 1 and 3: **1 unique order (J > N > H)**
- From Voter 2: **1 unique order (N > J > H)**
- From Voter 4: **1 unique order (J > H > N)**
- Voter 5: **does not introduce a new unique order since it fits elsewhere.**
- Voter 6: **may count as a distinct order but is not complete.**
Now, if we only consider the complete rankings:
1. J > N > H
2. N > J > H
3. J > H > N
Thus we have identified **3 unique preference orders** represented fully by voters.
Therefore, the correct answer is **three unique preference orders.**
- Voter 1: J > N > H
- Voter 2: N > J > H
- Voter 3: J > N > H
- Voter 4: J > H > N
- Voter 5: N > N > J
- Voter 6: H > H > H
Consolidating this information, we can identify the unique preference orders:
1. J > N > H (Voters 1 and 3)
2. N > J > H (Voter 2)
3. J > H > N (Voter 4)
4. N > J > H (Voter 5) - this is actually the same as preference order 2, so we don't count it again.
5. H > H > H (Voter 6) - this appears to be an invalid order because it doesn't rank all candidates, but we can treat it as a preference if needed.
Since J > N > H and N > J > H are counted only once, we have:
- From Voter 1 and 3: **1 unique order (J > N > H)**
- From Voter 2: **1 unique order (N > J > H)**
- From Voter 4: **1 unique order (J > H > N)**
- Voter 5: **does not introduce a new unique order since it fits elsewhere.**
- Voter 6: **may count as a distinct order but is not complete.**
Now, if we only consider the complete rankings:
1. J > N > H
2. N > J > H
3. J > H > N
Thus we have identified **3 unique preference orders** represented fully by voters.
Therefore, the correct answer is **three unique preference orders.**