Question

We have two bookshelves. let tn be the number of ways to first partition a set of n distinct books into two non-empty blocks, and then to line up one block on the top bookshelf and to line up the other block on the bottom bookshelf. find a closed formula for tn.

Answer 1

The closed formula for tn is tn = 2^n - 2, which accounts for the number of ways to partition n distinct books into two non-empty blocks and arrange them on two bookshelves.

Step-by-step explanation:

To find a closed formula for tn, which represents the number of ways to partition n distinct books into two non-empty blocks and then line them up on two bookshelves, we must consider the partitioning process. For any given book, there are two choices: it can either go on the top shelf or the bottom shelf. However, we must subtract the two cases where all books are on one shelf. Since each of the n books has 2 choices, there are 2^n ways to place the books. Subtracting the two invalid options gives us a total of 2^n - 2 arrangements. Therefore, tn = 2^n - 2.