Two combinatorics questions.

1. In an office, at various times during the day, the boss gives the secretary a letter to type, each time putting the letter on top of the pile in the secretary's in-box. When there is time, the secretary takes the top letter off the pile and types it. There are nine letters to be typed during the day and the boss delivers them in the order 1,2,3,4,5,6,7,8,9. While leaving for lunch, the secretary tells a colleague that letter 8 has already been typed, but says nothing else about the morning's typing. The colleague wonders which of the nine letters remain to be typed after lunch and in what order they will be typed. Based upon the information given, how many such after-lunch typing orders are possible? (There are no letters left to typed is one of the possibilities.)

2. Let $\displaystyle \mathbb{P}_n$ be the set of subsets of $\displaystyle \{1,2, \cdots, n\}$. Let $\displaystyle c(n,m)$ be the number of functions $\displaystyle f : \mathbb{P}_n \to \{1,2, \cdots, m\}$ such that $\displaystyle f(A \cap B) = \mbox{min}\{f(A), f(B)\}$. Prove that $\displaystyle c(n,m) = \sum_{j=1}^m j^n$.