Question

question 1
5 pts
number of voters: 8, 6, 2, 3, 5
1st choice: a, b, c, d, e
2nd choice: b, d, a, e, a
3rd choice: c, e, e, a, d
4th choice: d, c, b, c, b
5th choice: e, a, d, b, c
using the borda count method the winner of this election is:
options: a, b, e, d

Explanation:

Step1: Define Borda Points

In Borda Count, 1st choice gets 5 points, 2nd gets 4, 3rd gets 3, 4th gets 2, 5th gets 1.

Step2: Calculate Points for A

- 8 voters: 1st (5) + 2nd (4) + 3rd (3) + 4th (2) + 5th (1)? No, wait: For each group of voters, multiply their choice's rank points.
Group 1 (8 voters): A is 1st (5) → 8×5 = 40
Group 2 (6 voters): A is 5th (1) → 6×1 = 6
Group 3 (2 voters): A is 2nd (4) → 2×4 = 8
Group 4 (3 voters): A is 3rd (3) → 3×3 = 9
Group 5 (5 voters): A is 2nd (4) → 5×4 = 20
Total for A: 40 + 6 + 8 + 9 + 20 = 83

Step3: Calculate Points for B

- Group 1 (8): B is 2nd (4) → 8×4 = 32
- Group 2 (6): B is 5th (1) → 6×1 = 6
- Group 3 (2): B is 4th (2) → 2×2 = 4
- Group 4 (3): B is 5th (1) → 3×1 = 3
- Group 5 (5): B is 4th (2) → 5×2 = 10
Total for B: 32 + 6 + 4 + 3 + 10 = 55

Step4: Calculate Points for C

- Group 1 (8): C is 3rd (3) → 8×3 = 24
- Group 2 (6): C is 4th (2) → 6×2 = 12
- Group 3 (2): C is 5th (1) → 2×1 = 2
- Group 4 (3): C is 4th (2) → 3×2 = 6
- Group 5 (5): C is 5th (1) → 5×1 = 5
Total for C: 24 + 12 + 2 + 6 + 5 = 49

Step5: Calculate Points for D

- Group 1 (8): D is 4th (2) → 8×2 = 16
- Group 2 (6): D is 2nd (4) → 6×4 = 24
- Group 3 (2): D is 5th (1) → 2×1 = 2
- Group 4 (3): D is 4th (2) → 3×2 = 6
- Group 5 (5): D is 3rd (3) → 5×3 = 15
Total for D: 16 + 24 + 2 + 6 + 15 = 63

Step6: Calculate Points for E

- Group 1 (8): E is 5th (1) → 8×1 = 8
- Group 2 (6): E is 3rd (3) → 6×3 = 18
- Group 3 (2): E is 3rd (3) → 2×3 = 6
- Group 4 (3): E is 2nd (4) → 3×4 = 12
- Group 5 (5): E is 5th (1)? No, Group 5: E is... Wait, Group 5 (5 voters) 1st is E? No, Group 5 1st is E? Wait no, Group 5 1st choice is E? Wait the table: Group 5 (5 voters) 1st is E? Wait no, the table rows: 1st Choice column 5 is E. Wait, let's recheck Group 5 (5 voters):
1st: E (5) → 5×5 = 25
2nd: A (4) → 5×4 = 20 (already counted in A)
Wait no, for E:
Group 1 (8): E is 5th (1) → 8×1 = 8
Group 2 (6): E is 3rd (3) → 6×3 = 18
Group 3 (2): E is 3rd (3) → 2×3 = 6
Group 4 (3): E is 2nd (4) → 3×4 = 12
Group 5 (5): E is... Wait Group 5 1st choice is E (5), 2nd A (4), 3rd D (3), 4th B (2), 5th C (1). So for E in Group 5: 1st (5) → 5×5 = 25? Wait no, I messed up earlier. Oh no! I made a mistake in Step2 for A. Wait, no: For each candidate, in each group, their rank (1st=5, 2nd=4, 3rd=3, 4th=2, 5th=1). So for each group, for candidate X, find their rank in that group and multiply by number of voters.

Let's redo E properly:
- Group 1 (8 voters): E is 5th (1) → 8×1 = 8
- Group 2 (6 voters): E is 3rd (3) → 6×3 = 18
- Group 3 (2 voters): E is 3rd (3) → 2×3 = 6
- Group 4 (3 voters): E is 2nd (4) → 3×4 = 12
- Group 5 (5 voters): E is 1st (5) → 5×5 = 25
Total for E: 8 + 18 + 6 + 12 + 25 = 69

Wait, earlier A calculation was wrong. Let's redo A:

Step2 (Redo A):

- Group 1 (8): A is 1st (5) → 8×5 = 40
- Group 2 (6): A is 5th (1) → 6×1 = 6
- Group 3 (2): A is 2nd (4) → 2×4 = 8
- Group 4 (3): A is 3rd (3) → 3×3 = 9
- Group 5 (5): A is 2nd (4) → 5×4 = 20
Total: 40 + 6 + 8 + 9 + 20 = 83 (correct, because Group 5 2nd is A, so A's rank in Group 5 is 2nd (4), not 1st. 1st in Group 5 is E.

Now E: 8 + 18 + 6 + 12 + 25 = 69.

Now check D again:
- Group 1 (8): D is 4th (2) → 8×2 = 16
- Group 2 (6): D is 2nd (4) → 6×4 = 24
- Group 3 (2): D is 5th (1) → 2×1 = 2
- Group 4 (3): D is 4th (2) → 3×2 = 6
- Group 5 (5): D is 3rd (3) → 5×3 = 15
Total: 16 + 24 + 2 + 6 + 15 = 63 (correct).

B: 32 (Group1: B 2nd) + 6 (Group2: B 5th) + 4 (Group3: B 4th) + 3 (Group4: B 5th) + 10 (Group5: B 4th) → 32+6=38, +4=42, +3=45, +10=55 (correct).

C: 24 (Group1: C 3rd) + 12 (Group2: C 4th) + 2 (Group3: C 5th) + 6 (Group4: C 4th) + 5 (Group5: C 5th) → 24+12=36, +2=38, +6=44, +5=49 (correct).

Now A: 83, B:55, C:49, D:63, E:69. Wait, A has 83, which is higher than E (69) and D (63). Wait, but let's check again for A in Group 4 (3 voters): 1st Choice D, 2nd E, 3rd A, 4th C, 5th B. So A is 3rd (3) → 3×3=9 (correct). Group 5 (5 voters): 1st E, 2nd A, 3rd D, 4th B, 5th C. So A is 2nd (4) → 5×4=20 (correct). Group 3 (2 voters): 1st C, 2nd A, 3rd E, 4th B, 5th D. So A is 2nd (4) → 2×4=8 (correct). Group 2 (6 voters): 1st B, 2nd D, 3rd E, 4th C, 5th A. So A is 5th (1) → 6×1=6 (correct). Group 1 (8 voters): 1st A, 2nd B, 3rd C, 4th D, 5th E. So A is 1st (5) → 8×5=40 (correct). Total 40+6+8+9+20=83.

E: Group 1 (8): 5th (1) → 8×1=8; Group 2 (6): 3rd (3) → 6×3=18; Group 3 (2): 3rd (3) → 2×3=6; Group 4 (3): 2nd (4) → 3×4=12; Group 5 (5): 1st (5) → 5×5=25. 8+18=26, +6=32, +12=44, +25=69. Correct.

So A has 83, which is the highest. Wait, but let me check once more. Maybe I made a mistake in E's Group 5. Group 5 (5 voters) 1st choice is E, so E gets 5 points per voter here: 5×5=25. Correct.

So A's total is 83, which is higher than others.

Answer:

A. A