Four candidates, Harrison (H), Lennon (L), McCartney (M), and Starr (S), are running for regional manager. After the polls close, ranked ballots are tallied, and the results are summarized in the table.Number of Ballots Cast27 97 47 83 32 16 241st choice H L L M M S S2nd choice L M S L L H M3rd choice S H M H S L H4th choice M S H S H M L(a) How many votes were cast? votes(b) Use the plurality method of voting to determine the winner.HarrisonLennon    McCartneyStarr(c) What percent of the votes did the winner in part (b) receive? (Round your answer to the nearest whole number.) %(d) Use the instant runoff method to determine the winner.HarrisonLennon    McCartneyStarr(e) What percent of the votes did the winner in part (d) receive? (Round your answer to the nearest whole number.) %(f) Use the Borda count method to determine the winner.HarrisonLennon    McCartneyStarr(g) How many points did the winner in part (f) receive? points(h) Use the pairwise comparison method to determine the winner.HarrisonLennon    McCartneyStarr(i) How many points did the winner in part (h) receive? points

(a) To find the total number of votes cast, sum the number of ballots in each column:

27 + 97 + 47 + 83 + 32 + 16 + 24 = 326 votes

(b) Using the plurality method, we count only the first-choice votes for each candidate:

• Harrison (H): 27
• Lennon (L): 97 + 47 = 144
• McCartney (M): 83 + 32 = 115
• Starr (S): 16 + 24 = 40

The candidate with the most first-choice votes is Lennon (L) with 144 votes.

(c) To find the percentage of votes the winner received:

(144 / 326) * 100 ≈ 44%

(d) Using the instant runoff method, we eliminate the candidate with the fewest first-choice votes and redistribute those votes according to the next preferences until one candidate has a majority.

First round:

• Harrison (H): 27
• Lennon (L): 144
• McCartney (M): 115
• Starr (S): 40

Starr (S) is eliminated. Redistribute Starr's votes:

• 16 votes go to Harrison (H)
• 24 votes go to McCartney (M)

Second round:

• Harrison (H): 27 + 16 = 43
• Lennon (L): 144
• McCartney (M): 115 + 24 = 139

Harrison (H) is eliminated. Redistribute Harrison's votes:

• 27 votes go to Lennon (L)
• 16 votes go to McCartney (M)

Final round:

• Lennon (L): 144 + 27 = 171
• McCartney (M): 139 + 16 = 155

Lennon (L) wins with 171 votes.

(e) To find the percentage of votes the winner received in the instant runoff:

(171 / 326) * 100 ≈ 52%

(f) Using the Borda count method, assign points based on ranking positions (4 points for 1st choice, 3 points for 2nd choice, 2 points for 3rd choice, 1 point for 4th choice):

• Harrison (H): (27 * 4) + (97 * 3) + (47 * 2) + (83 * 1) = 108 + 291 + 94 + 83 = 576
• Lennon (L): (97 * 4) + (47 * 4) + (83 * 3) + (32 * 2) + (16 * 1) = 388 + 188 + 249 + 64 + 16 = 905
• McCartney (M): (83 * 4) + (32 * 4) + (16 * 3) + (24 * 2) + (27 * 1) = 332 + 128 + 48 + 48 + 27 = 583
• Starr (S): (16 * 4) + (24 * 4) + (27 * 3) + (97 * 2) + (47 * 1) = 64 + 96 + 81 + 194 + 47 = 482

The candidate with the most points is Lennon (L) with 905 points.

(g) The winner in part (f) received 905 points.

(h) Using the pairwise comparison method, compare each candidate against each other:

• Harrison vs. Lennon: Lennon wins
• Harrison vs. McCartney: McCartney wins
• Harrison vs. Starr: Harrison wins
• Lennon vs. McCartney: Lennon wins
• Lennon vs. Starr: Lennon wins
• McCartney vs. Starr: McCartney wins

Points:

• Harrison: 1
• Lennon: 3
• McCartney: 2
• Starr: 0

The candidate with the most points is Lennon (L) with 3 points.

(i) The winner in part (h) received 3 points.