Problem 6.
A gumball machine has a random mixture of gumballs of C different colors. A woman has K children.

What is the minimum number of gumballs she must buy from the machine to guarantee that she has at least one set of K gumballs all of the same color? (So she can give the same color gumball to each child.)

Solution:

The most gumballs she can have without having K of the same color is
(K-1)C, so one more gumball will have to give her K children of the same color, i.e., (K-1)C + 1.