Twelve people serve on a board and are considering three alternatives: A, B, and C. The choices followed by vote is shown in following table. Determine the winner, if any, using a Borda count.
(ABC) 2
(ACB) 4
(BAC) 2
(BCA) 1
(CAB) 2
(CBA) 1
The winner is _____?
Well, let's do some math to determine the winner with the Borda count. For each position a choice receives, it gets a certain number of points. The first choice gets 3 points, the second choice gets 2 points, and the third choice gets 1 point.
Now, let's tally up those points for each choice.
Choice A: (2 * 1) + (4 * 2) + (2 * 3) = 14 points
Choice B: (2 * 2) + (1 * 1) + (2 * 2) = 10 points
Choice C: (2 * 3) + (1 * 3) + (4 * 1) = 13 points
Ah, it seems like Choice A has the most points with a grand total of 14 points! So, according to the Borda count, the winner is Choice A. But please don't be disappointed, Choices B and C, you did your best! Time for some celebratory confetti! 🎉🎊🎉
To determine the winner using the Borda count method, we assign points to each alternative based on the preferences indicated in the table. Each alternative will receive points based on its position in each vote.
For example, in the first vote (ABC), alternative A is in the first position, so it receives 2 points. Alternative B is in the second position, so it receives 1 point. Alternative C is in the last position, so it receives 0 points.
By calculating the Borda count for each alternative in each vote and summing up the points, we can determine the winner.
Let's calculate the Borda count for each alternative:
Alternative A:
(ABC) 2 + (ACB) 4 + (BAC) 2 + (BCA) 1 + (CAB) 2 + (CBA) 1 = 12 points
Alternative B:
(ABC) 0 + (ACB) 0 + (BAC) 1 + (BCA) 2 + (CAB) 1 + (CBA) 2 = 6 points
Alternative C:
(ABC) 1 + (ACB) 2 + (BAC) 0 + (BCA) 0 + (CAB) 0 + (CBA) 0 = 3 points
From the calculation, we can see that alternative A received the highest Borda count of 12 points. Therefore, the winner according to the Borda count method is alternative A.
To determine the winner using the Borda count method, we assign points to each alternative based on their rankings in each vote. The alternative with the highest total points will be the winner.
In the Borda count method, the first alternative in each vote receives (n-1) points, the second alternative receives (n-2) points, and so on, where n is the number of alternatives being considered.
Let's calculate the points for each alternative:
For alternative A:
(ABC) - A receives 2 points because it is ranked first.
(BAC) - A receives 2 points because it is ranked second.
(CAB) - A receives 1 point because it is ranked third.
Total points for A = 2 + 2 + 1 = 5
For alternative B:
(ACB) - B receives 2 points because it is ranked second.
(BCA) - B receives 1 point because it is ranked third.
(CBA) - B receives 1 point because it is ranked second.
Total points for B = 2 + 1 + 1 = 4
For alternative C:
(ABC) - C receives 0 points because it is ranked third.
(ACB) - C receives 1 point because it is ranked third.
(BCA) - C receives 2 points because it is ranked first.
(CAB) - C receives 2 points because it is ranked second.
(CBA) - C receives 0 points because it is ranked third.
Total points for C = 0 + 1 + 2 + 2 + 0 = 5
From the calculations above, both alternatives A and C have the highest total points with a score of 5 each. Therefore, there is a tie between alternatives A and C as the winners with the highest Borda count.