 a single to equiprobable outcomes, even over a number of trials as high as 1000 and after trying different tossing methods. If the coin is not 100% balanced between the two sides in terms of mass, nonuniform PDFs can be obtained. 1.2 Combinatorics choice left, which gives for the total number of possibilities: Q=10x9x8x7x 6x5x4 x3x2x1. By definition, this product is called the factorial of 10 and is noted 10! More generally, for any integer number n we have the factorial definition: !=nx(n-1)x(n-2) X x3x2x1. (1.6) 1!=1, 2!=2x1=2, 3!=3x2x1=6, and SO on. One can easily compute 10! (using, for instance, the factorial function of a scientific pocket calculator) to find 10! = 362880 3.6 million. There is an impressive number of ways indeed to personalize one's office shelves! When it comes to somewhat greater arguments (e.g., n = 100), the factorials cannot be computed by hand or through a pocket calculator, owing to their huge size. To provide an idea (as computed through a spreadsheet math function up to a maximum of 170!): 50! = 3.0 x 1064, 100! = 9.3 x 10 57, 1701 ~ 7.2 x 10 306. In this case it is possible to use Stirling's approximation theorem, which is written as n!~n"e-"V2rn. (1.7) Remarkably, the Stirling theorem is accurate within 1.0% for arguments n> 10 and within 0.20% for arguments n 40. To summarize, the factorial of n is the number of ways to arrange n distinguishable elements into a given orderly fashion. Such an arrangement is also called a permutation. What about the factorial of the number zero? By convention, mathematicians have set the odd property 0!=1. We shall accept here that 0!= 1 without advanced justification. Simply put, if not a satisfactory explanation, there is only one way to arrange/permute a set containing zero element.³ Consider next a second example. For the math-oriented reader, it is interesting to mention that the factorial function can be generalized to any real or even complex arguments x, i.e., x! = r(x + 1), where is the gamma function, which can be computed numerically. 