creasingly accurate as Q becomes larger. The expression "should be observed" progresses towards "should be absolutely certain." To top off this statement, we can say that an event with a finite, nonzero probability is absolutely certain to occur at least once in the unbounded future. Such a statement is true provided the physics governing the event (and its associated PDF) remain indefinitely the same, or is "invariant by time translation" in physicists' jargon. Combinatorics The examples in the previous section concerned events whose numbers of possible occurrences in a given trial are easily defined. For instance, we know for certain that a single die has exactly six faces, with their corresponding numbers of spots. But when adding spots from two dice (rolling dice, game 2), we had to go through a kind of inventory in order to figure out all the different possibilities. While this inventory was straightforward in this example, in the general case it can be much more tedious if not very complex. Studying the number of different ways of counting and arranging events is called combinatorial analysis or combinatorics. Instead of formalizing combinatorics at once, a few practical examples are going to be used to introduce its underlying concepts progressively. Arranging books on a shelf This is a recurrent problem when moving into a new home or office. A lazy solution is to unpack the books and arrange them on the shelf from left to right, as we randomly pick them up from the box. How many different ways can we do this? Answer: Say the shelf can hold ten books, and number the book position from the left. Assume that the box fits in ten different (or distinguishable) books. The number of ways to pick up the first book to place in position 1 is, therefore, Q = 10. For the second book to put in position 2, there remain nine possibilities. So far, there have been Q = 10 x 9 possibilities. Positioning next the third book, the count of possibilities becomes = 10 X 8. And SO on, until the last book, for which there is only