One can imagine an embellishment to the general pigeon-hole picture to make the principle apply. Imagine 12 pigeon-holes, and that we are enforcing these rules: you use empty spots first when they are available; you can only put red balls in the first ten slots; you can only put blue balls in the last two slots.
The pigeon-hole principal says that if 13 balls are sorted into these pigeon-holes, that at least one of the slots has two balls in it. If one of the white slots has 2 balls that means, according to our rules, that there are at least 11 white balls (a contradiction to the limited supply of white balls.) Since that cannot happen, there must be a spot with two blue balls. Again according to our rules, that means we filled the two blue slots and then added another blue. So, there must be at least three blue balls.
To show this is minimal, you just point out that selecting 10 whites and 2 blues shows that 12 is not sufficient.