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At a recently held election one party fielded 3 candidates A,B and C. In this election people can vote for any number of candidates in a desired party. The voting pattern of 100 people as follows: The 3 candidates; A,B and C received $90, 60$ and $57$ votes respectively while 5 people did not vote for anyone. What is the most number of people who could have voted for all 3?

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up vote 14 down vote accepted

The answer is

56

Explanation:

57 is obviously a hard maximum, since that is the number of votes received by candidate C. But if 57 people voted for all 3, then the maximum number of total voters is 93: 57 for all 3, 3 for B, and 33 for A.

Next consider the case of 56 voting for all 3. This results in 1 for C, 4 for B, and 34 for A, for a total of 95 voters. (5 people abstained.)

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