The Library of Babel and the digits of Pi

One of the most famous stories by Jorge Luis Borges is The Library of Babel (La Biblioteca de Babel). This library contains all possible books. For this statement to make sense, we must specify the definition of a book. For Borges, each book is made of one million three hundred and twelve thousand characters, chosen among all the possible permutations of that length and built with a set of 25 basic characters (the space, 22 letters, and two punctuation marks).

The number of books in the Library is huge, for the number of permutations of a certain number of symbols grows according to the factorial of their length. The factorial of a number N is obtained by multiplying together all the natural numbers from 1 to N:

N!=1×2×3×...×N

The result of this operation grows disproportionately. Thus, 5! = 120; 10! = 3,628,800; 100!>9×10¹⁵⁷. The number of books in the Library of Babel does not grow so quickly, because each book contains many repetitions of the symbols, which decreases the number of possibilities, but consider what will be the factorial of 1,312,000, if the factorial of 100 is a number with 157 digits.

The number of books in the Library of Babel is huge, but it is not infinite. That set of books contains all the possible chains of that length, whether or not they make sense. But that means that it contains all the books that have ever been written; all that could ever be written; all translations of each book to all the existing or possible languages​​...