In an election there were five constituencies S1, S2, S3, S4, and S5 with 20 voters each all of whom voted. Three parties A, B, and C contested the elections. The party that gets the maximum number of votes in a constituency wins that seat. In every constituency there was a clear winner.
The following additional information is available:
- Total number of votes obtained by A, B, and C across all constituencies are 49, 35 and 16 respectively. - S2 and S3 were won by C while A won only S1. - Number of votes obtained by B in S1, S2, S3, S4, and S5 are distinct natural numbers in increasing order.
Solution
Since each constituency has a clear winner and the total number of votes polled in each constituency is 20, we can conclude that the minimum number of votes the winner party can get is 8. If a party receives fewer than 8 votes, it cannot be the winner in any case. We can find this value by assigning equal number of votes to the two losing parties and the remaining votes to the winning one.
Using this minimum value for votes of the winning party, we can conclude that Party C, which won exactly two constituencies, and polled a total of 16 votes, has to have gotten 8 votes each in the winning constituencies and 0 votes in all the others.
Further, since all the votes in constituencies S4 and S5 go to A and B only (C having gotten 0 in each), and since B has won S4 and S5, B has to have gotten more than half (which is 10) votes in constituencies S4 and S5.
In S2 and S3, where C won with 8 votes each, the remaining 12 votes can be distributed in two ways: (5,7) and (6,6), as both A and B must receive fewer votes than C. Also, B has to have increasing values in S2 to S3, so B can have (5,7), or (5,6), or (6,7) in S2 and S3 respectively. In the case where Party B gets (5,7) in S2 and S3 respectively, 35-12= 23 will be the sum for S1, S4, and S5 for B. Since B gets more than 10 votes in S4 and S5, the only arrangement for B from S1 to S5 in this case is (0,5,7,11,12), which is not possible as all constituency votes for B have to be natural numbers. Similarly, if B gets (5,6) in S2 and S3 respectively, the remaining sum will be 35-11=24 for S1, S4, and S5, this gives two possibilities: either (1,5,6,11,12), which is possible, or (0,5,6,11,13), which is not possible as all constituency votes for Party B have to be natural numbers. Finally, since 11+12=23 is the minimum possible sum for B in S4 and S5, and since (6,7) in S2 and S3 has sum 6+7=13, 13+23= 36 exceeds 35 and the case of (6,7) votes for B in S2 and S3 is not possible. We are therefore left with the following possible cases. The votes for A are simply calculated by subtracting the assigned votes from the total votes in each constituency (such that they also give a sum of 49).
Based on the table above, it can be concluded that the constituency in which B got lower number of votes compared to A and C is S2.
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