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Eight friends consider and vote for the five shops they want to go to. However there is no absolute winner, and shops chosen by less than two friends of the group are not considered before they vote again.

a) Explain the different ways the votes could have been allocated among the shops, so that there was no absolute winner?

b) Explain why at least 1 of the shops achieved at least 3 votes in the first round of voting, if 3 were not considered after the first vote?

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Are you told how many friends are in the group? – Brian M. Scott Jan 9 '13 at 16:47
There are 8 friends – DeeDee Jan 9 '13 at 17:16
up vote 1 down vote accepted

Let’s look at (b) first. Suppose that $3$ shops were not considered after the first round; then $3$ shops received at most $1$ vote each, leaving at least $5$ votes that must have gone to the other two shops. No matter how you split $5$ votes between $2$ shops, at least one of the shops must get at least $3$ votes: if each of them got $2$ or fewer, that would be a total of at most $4$.

Now for (a). If a shop gets $5$ or more votes in the first round, it must be the absolute winner: no other shop could get more than $3$ votes. What if a shop gets $4$ votes? It will be the absolute winner unless another shop also gets $4$ votes. Thus, one way to have no absolute winner is to have the $8$ votes evenly split between two shops.

What if the highest number of votes received by any shop is $3$? If that shop is not to be the absolute winner, there must be another shop with $3$ votes, and the other $2$ votes can either go to a third shop or be split between two of the other shops. In other words, the votes must split $3,3,2$ or $3,3,1,1$. I’ll leave the last little bit to you: is it possible to have no absolute winner when the highest number of votes received by any shop is $2$? What about $1$?

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