1. At the ice cream store
The Baskin-Baskin ice cream store has the unusual slogan "exactly 4n flavor combinations." By this they mean that if you are willing to spend n dollars for ice cream (plus a small fee for the cone), you can choose exactly 4n combinations of ice cream scoops on a two-scoop cone, where different orders are considered to be different combinations. This is a great improvement on the Robbins-Robbins store down the street, where they provide only vanilla scoops for $0 and chocolate scoops for $1, giving exactly one $0 cone (vanilla-vanilla), two $1 cones (vanilla-chocolate and chocolate-vanilla) and one $2 cone (chocolate-chocolate).
Deduce from Baskin-Baskin's slogan a closed-form expression for exactly how many different scoops of ice cream cost $n, for each value of n.
2. A tricky sum
Find a closed-form expression in a and n for the value of
3. Counting change
Suppose you have an infinite collection of $1 bills (1), $1 coins (①) and $2 bills (2). Find a closed-form expression in n for the number of distinct ways to make $2n. (Example: for n=1, we get 4 ways to make $2: 11, 1①, ①①, and 2.)
Let and A and B be independent events on some probability space. Prove or disprove: their complements -A and -B are also independent.