A specific board of supervisors uses the weighted voting system \( [54: 25,23,19,12,9,1] \). Assume that the players are \( \mathrm{P}_{1} \) through \( \mathrm{P}_{6} \). Complete parts (a) through (d) below. (c) List all the five-player winning coalitions and find the critical players in each coalition. A. \( \left\{P_{1}, P_{2}, P_{3}, P_{4}, P_{5}\right\},\left\{P_{1}, P_{2}, P_{3}, P_{4}, P_{6}\right\},\left\{P_{1}, P_{2}, P_{3}, P_{5}, P_{6}\right\},\left\{P_{1}, P_{2}, P_{4}, P_{5}, P_{6}\right\} \), \[ \left\{P_{1}, P_{3}, P_{4}, P_{5}, P_{6}\right\},\left\{P_{2}, P_{3}, P_{4}, P_{5}, P_{6}\right\} \] B. \( \left\{P_{1}, P_{2}, P_{3}, P_{4}, P_{5}\right\},\left\{P_{1}, P_{2}, P_{3}, P_{4}, P_{6}\right\},\left\{P_{1}, P_{2}, P_{3}, P_{5}, P_{6}\right\},\left\{P_{1}, P_{2}, P_{4}, P_{5}, P_{6}\right\} \), \[ \left\{P_{1}, P_{3}, P_{4}, P_{5}, P_{6}\right\},\left\{\overline{P_{2}}, P_{3}, P_{4}, P_{5}, P_{6}\right\} \] C. \( \left\{P_{1}, P_{2}, P_{3}, P_{4}, P_{5}\right\},\left\{P_{1}, P_{2}, P_{3}, P_{4}, P_{6}\right\},\left\{P_{1}, P_{2}, P_{3}, P_{5}, P_{6}\right\},\left\{P_{1}, P_{2}, P_{4}, P_{5}, P_{6}\right\} \), \[ \left\{\overline{P_{1}}, \overline{P_{3}}, P_{4}, P_{5}, P_{6}\right\},\left\{\overline{P_{2}}, \overline{P_{3}}, P_{4}, P_{5}, P_{6}\right\} \] D. \( \left\{P_{1}, P_{2}, P_{3}, P_{4}, P_{5}\right\} \cdot\left\{P_{1}, P_{2}, P_{3}, P_{4}, P_{6}\right\},\left\{P_{1}, P_{2}, P_{3}, P_{5}, P_{6}\right\},\left\{P_{1}, P_{2}, P_{4}, P_{5}, P_{6}\right\} \), \( \left\{\overline{P_{1}}, P_{3}, P_{4}, P_{5}, P_{6}\right\} \cdot\left\{\overline{P_{2}}, \overline{P_{3}}, P_{4}, P_{5}, P_{6}\right\} \)

To find the five-player winning coalitions and their critical players, let's start by calculating the total weight of votes for each player according to the given weighted voting system \( [54: 25, 23, 19, 12, 9, 1] \): - \( P_1 \): 25 votes - \( P_2 \): 23 votes - \( P_3 \): 19 votes - \( P_4 \): 12 votes - \( P_5 \): 9 votes - \( P_6 \): 1 vote The total weight required to win a vote is 54 votes. Now, let's list the five-player winning coalitions and determine their critical players. ### Winning Coalitions and Critical Players: 1. \( \{P_1, P_2, P_3, P_4, P_5\} \) - Total Weight: 25 + 23 + 19 + 12 + 9 = 88 (winning coalition) - Critical Players: \( P_1, P_2, P_3, P_4, P_5 \) (removing any player results in less than 54) 2. \( \{P_1, P_2, P_3, P_4, P_6\} \) - Total Weight: 25 + 23 + 19 + 12 + 1 = 80 (winning coalition) - Critical Players: \( P_1, P_2, P_3, P_4 \) 3. \( \{P_1, P_2, P_3, P_5, P_6\} \) - Total Weight: 25 + 23 + 19 + 9 + 1 = 77 (winning coalition) - Critical Players: \( P_1, P_2, P_3, P_5 \) 4. \( \{P_1, P_2, P_4, P_5, P_6\} \) - Total Weight: 25 + 23 + 12 + 9 + 1 = 70 (winning coalition) - Critical Players: \( P_1, P_2, P_4 \) 5. \( \{P_1, P_3, P_4, P_5, P_6\} \) - Total Weight: 25 + 19 + 12 + 9 + 1 = 66 (winning coalition) - Critical Players: \( P_1, P_3, P_4 \) 6. \( \{P_2, P_3, P_4, P_5, P_6\} \) - Total Weight: 23 + 19 + 12 + 9 + 1 = 64 (winning coalition) - Critical Players: \( P_2, P_3, P_4 \) ### Conclusion: The winning coalitions consist of several combinations of players with the total weight exceeding the threshold of 54. Each coalition can have different critical players whose votes are essential for maintaining that winning status. Remember, a critical player is one that, if removed from the coalition, the coalition would no longer meet the required majority.