Suppose there are five voters, A, B, C, D, and E, with 7, 6, 5, 4, and 3 votes, respectively. Suppose that a simple majority is required to win. How many winning coalitions are there?
A)31
B)17
C)13
D)8

Question

Quizplus · Accepted Answer

For a coalition to win, it needs at least 4 votes. There are 5 individual voters, and only one coalition with all of them would achieve the majority.

There are also combinations of 4 voters, which are ${5 \choose 4} = 5$  possible winning coalitions.

Lastly, there are no combinations of 3 voters that constitute a winning coalition because the maximum number of votes that 3 voters can have is 18, which is not a majority.

Therefore, there are a total of 5 + 5 + 1 = 11 winning coalitions. However, we also need to consider the possibility of a tie, which occurs if two coalitions have exactly the same number of votes.

The only way this can happen is if the two smallest voters, D and E, are split between two coalitions. There are two possible ways this can happen:

1. D and E both join a coalition with A, B, or C. In this case, there are ${3 \choose 2} = 3$ possible tie-breaking coalitions. 
2. D and E are on opposite sides of the split, with one joining a coalition with A, B, and C, and the other joining a coalition with just A and B or just B and C. There are two such possible pairs, and for each pair there are 2 possible tie-breaking coalitions.

Therefore, there are a total of 11 + 3 + 2(2) = 18 winning coalitions, which is answer choice (B). However, note that the question asks for the number of winning coalitions, not including ties. In this case, the answer is 13, which is answer choice (C).

Suppose There Are Five Voters, A, B, C, D, and E