# 1. Presidents and vice presidents

Show that, for all nāā,

Hint: What does each side of the equation count?

# 2. Mystery chocolates

A box of n chocolates contains m in flavors that you don't like. Assuming that you choose r of the chocolates to eat uniformly at random, what is the probability that at least one of them is a flavor that you don't like?

# 3. Monochrome sets

Suppose we have a set S of n elements, and we assign each element one of r colors. Call a subset A of S **monochrome** if every element of A is assigned the same color. Given n, r, and k, show that there exists a coloring of S in which at most subsets A of size k are monochrome.

Hint: On average, how many subsets are monochrome in a random coloring?