I am reading about the Pigeonhole principle and the following problem under that section:
A state has $7$ counties. In one year, three candidates run in a statewide election. Is it possible that in each county the same number of people vote and the candidate who gets the highest number of votes does not get the highest number in any county?
My first thought was that the three candidates $c$ are the "boxes" and the votes $v$ are the "balls" according to the Pigeonhole principle. We have $v \gt rc \implies$ at least one candidate will get at least $r + 1$ votes (generalized version). But then I got stuck how to expand this.
I came up with an example that actually shows that what the question asks is possible:
| County-1 | County-2 | County-3 | County-4 | County-5 | County-6 | County-7 |
|---|---|---|---|---|---|---|
| A | A | B | A | C | A | C |
| B | C | B | A | C | B | C |
| B | C | C | B | B | B | B |
| A | A | C | B | B | A | B |
Here we see that $B$ has the most votes even though B is not the most voted in any county. I do not consider a tie as the same as the case of being the most voted person, but in any case it is not that important to my question.
When I checked the solution provided for the problem it actually did the same thing.
Book's solution:
If we have $100$ people voting we can have $A$ having $40$ votes in each county. Candidate $B$ having $50$ votes in three counties and $10$ votes in the remaining four counties while the candidate $C$ gets $10$ votes in the first three counties, and $50$ in the remaining four counties. This means that statewide, the candidate $A$ $280$ votes, candidate $B$ gets $190$ votes and candidate $C$ gets $230$ votes.
At this point, I am confused. How is the pigeonhole principle applicable here then if the actual book's solution offers just a counterexample?
What am I misunderstanding here?