A two-day mathematics conference has n participants. Some of the participants give a talk on Saturday, some others give a talk on Sunday. Nobody gives more than one talk, and there may be some people who do not give a talk at all. At the end of the conference, a few talks are selected to be included in a book. In how many different ways is this all possible if we assume that there is at least one talk selected for inclusion in the book?
I suppose that if A gives a talk on Saturday that is selected for the book, it is considered a different possibility than if A gives a talk on Sunday that is selected for the book.
So for n = 1, we would have 2 possibilities.
For n=2,
A speaks on Saturday (1 possibility)
B speaks on Saturday (1 possibility)
A speaks on Sunday (1 possibility)
B speaks on Sunday (1 possibility)
A and B speak on Saturday (3 possibilities)
A and B speak on Sunday (3 possibilities)
A speaks on Saturday and B speaks on Sunday (3 possibilities)
B speaks on Saturday and A speaks on Saturday (3 possibilities)
In all, there are 16 possibilities.
So, we need to find the number of possibilities as a function of n.