Why Is 0! = 1 ?

Consider n unique books arranged on a bookshelf. To arrange the books on this bookshelf, there are n books that can go in position 0, on the very left of the shelf. In the next position, directly to the right of position 0, position 1, 1 of the n-1 remaining books can be placed. In the next position, position 2, 1 of the remaining n-2 books can be placed.

Following this pattern, it`s clear that there is a choice of n-k books to go in position k. The number of ways of arranging the bookshelf overall is then n*(n-1)*(n-2)*...*2*1 = n! . There are n! ways of arranging n unique books on a bookshelf.

Test: let there be 3 unique books: Animal Farm ("A"), The Bible ("B"), and The Catcher in the Rye ("C"). There are 6 different ways of arranging these on the bookshelf: "ABC", "ACB", "BAC", "BCA", "CAB", "CBA". 3! = 3*2*1 = 6, which is consistent with the true result.

What about if there are 2 unique books? Then there are 2 ways of arranging these ("AB" and "BA") and 2!=2. What about 1 book? Only way to arrange this is "A" and 1!=1. What about 0 books? There is again only 1 way to arrange this: " ", where the bookshelf is empty. Therefore 0!=1.