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An employer surveys employees to see which lunch choice they should have at a work picnic. Here are the results:

\begin{tabular}{|l|l|l|l|l|}
\hline
& 8 voters & 6 voters & 3 voters & 3 voters \\
\hline
First choice & Tacos & Pizza & Sandwiches & Pizza \\
\hline
Second choice & Pizza & Tacos & Tacos & Sandwiches \\
\hline
Third choice & Sandwiches & Sandwiches & Pizza & Tacos \\
\hline
\end{tabular}

Based on these results, how many first-place votes are needed to be the winner using the instant runoff method?

A. 10
B. 11
C. Pizza wins. Tacos are eliminated in the first round, so pizza moves up in the first column.
D. 3
E. 2


Sagot :

Based on the results provided in the survey:

Here is a tabulated summary:
[tex]\[ \begin{array}{|l|c|c|c|c|} \hline & 8 \text{ voters} & 6 \text{ voters} & 3 \text{ voters} & 3 \text{ voters} \\ \hline \text{First choice} & \text{Tacos} & \text{Pizza} & \text{Sandwiches} & \text{Pizza} \\ \hline \text{Second choice} & \text{Pizza} & \text{Tacos} & \text{Tacos} & \text{Sandwiches} \\ \hline \text{Third choice} & \text{Sandwiches} & \text{Sandwiches} & \text{Pizza} & \text{Tacos} \\ \hline \end{array} \][/tex]

Step-by-Step Solution for Instant Runoff Voting:

1. First Choice Count:
- Tacos: 8 voters
- Pizza: 6 voters + 3 voters = 9 voters
- Sandwiches: 3 voters

2. Determine if there is a majority winner:
- Total voters = 8 + 6 + 3 + 3 = 20.
- A majority requires more than half of the total votes, so more than 10 votes are needed to win outright.

3. Check for a majority:
- Tacos: 8 votes
- Pizza: 9 votes
- Sandwiches: 3 votes

No single candidate has more than 10 votes, so we proceed to eliminate the candidate with the fewest votes.

4. Eliminate the candidate with the fewest votes:
- Sandwiches have the fewest votes (3 votes), so they are eliminated.

5. Redistribute votes based on the next choice for those who voted for Sandwiches:
- The 3 voters who initially voted for Sandwiches will have their votes transferred to their second choice:
- Second choice of Sandwiches voters: All 3 voters prefer Tacos as their second choice.

6. Recount with redistributed votes:
- Tacos: 8 original + 3 redistributed = 11 votes
- Pizza: 9 votes

7. Determine if there is a majority winner after redistribution:
- Tacos: 11 votes
- Pizza: 9 votes

With Tacos securing 11 votes, which is more than 10 and thus a majority, Tacos is the winner according to the instant runoff method.

Therefore, the number of first-place votes needed to be the winner using the instant runoff method, given the provided voting results, is at least: 11 votes.

And based on the process, Tacos wins.
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