Combinatorics question See attached image We have two booksh

Combinatorics question. See attached image.

We have two bookshelves. Let t be the number of ways to first partition a set of n distinct books into 2 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 close formula for tn

Solution

Consider in n books k books are arranged in 1 Shell then n-k books are in Shell 2

K books are arranged in k! Ways

K-1 books are arranged in (n-k)! Ways

These two Shell can be interchange in 2 ways

Total no.of ways is 2*k!*(n-k)!